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FIRST PRINCIPLES 


DIFFERENTIAL AND INTEGRAL 


CALCULUS, 


OR THE 


DOCTRINE OF FLUXIONS, 


INTENDED 


AS AN INTRODUCTION TO THE PHYSICO-MATHEMATICAL SCIENCES ; 


TAKEN CHIEFLY 
FROM THE MATHEMATICS OF BEZOUT, 


And translated from the French 


FOR THE USE OF THE STUDENTS OF THE UNIVERSITY AT 


CAMBRIDGE, NEW ENGLAND. 


= at . 
eg Ae neme oe) 
a ; vt 


CAMBRIDGE, N. E. 
PRINTED BY HILLIARD AND METCALF, 
At the University Press. 


SOLD BY W. HILLIARD, CAMBRIDGE, AND BY CUMMINGS, HILLIARD, & co 
NO. I CORNHILL, BOSTON. 


1824, 


DISTRICT OF MASSACHUSETTS, to wit: 
District Clerk’s Office. 


BE IT REMEMBERED, that on the ninth day of July A. D. 1824, in the forty-ninth year of 
the Independence of the United States of America, Cummings, Hilliard, & Co, of the said district 
have deposited in this office the title of a book, the right whereof they claim as proprictors, in 
the words following, to wit: 

“ First Principles of the Differential and Integral Calculus, or the Doctrine of Fluxions, in- 
tended as an Introduction to the Physico-Mathematical Sciences ; taken chiefly from the Mathe- 
matics of Bézout, and translated from the French for the use of the Students of the University at 
Cambridge, New-England.” 

In conformity to the.act of the Congress of the United States, entitled ** An act for the encour- 
agement of learning, by securing the copies of maps, charts, and books, to the authors and pro- 
prietors of such copies, during the times therein mentioned ;” and also to an act, entitled 
** An act supplementary to an act, entitled ‘An act for the encouragement of learning, by 
securing the copies of maps, charts, and books, to the authors and proprietors of such copies, 
during the times therein mentioned ;’ and extending the benefits thereof to the arts of design- 
ing, engraving, aud etching historical and other prints.” ; 

. JOHN W. DAVIS, 


i Clerk of the District of Massachusetts. 


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ADVERTISEMENT. 


Tue following treatise, except the introduction and notes, is a 
translation of the Principes de Calcul qui servent ad’ Introduction 
aux Sciences Physico-Mathématiques of Bezout. It was selected 
on account of the plain and perspicuous manner for which the 
author is so well known, as also on account of its brevity and 
adaptation in other respects to the wants of those who have but 
little time to devote to such studies. The easier and more im- 
portant parts are distinguished from those which are more difficult 
or of less frequent use, by being printed in a larger character. 
In the Introduction, taken from Carnot’s Reflexions sur la Met- 
aphysique du Calcul Infimtesimal, a few examples are given to 
show the truth of the infinitesimal method, independently of its 
technical form. Moreover in the 4th of the notes, subjoined at 
the end, some account is given from the same work, of the 
methods previously in use, analogous to the Infinitesimal Analy- 


sis. The other notes are intended to supply the deficiencies o: 
Lacroix’s Algebra (Cambridge Translation), considered as a 


preparatory work. 

Since this treatise was announced, the compiler of the Cam- 
bridge Mathematics has been obliged, on account of absence from 
the country and infirmity of sight, to resign his work into other 
hands. This circumstance is mentioned to account for the delay 
attending the publication, as well as the occasional want of confor- 
mity to other parts of the course in the mode of rendering certain 
words and phrases which a revision of the translation, had it been 
practicable, would have easily remedied. 


Cambridge, July 1824. 


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Sian ¥ 


b SS OR”, 
butte 


oe 


CONTENTS. 


INTRODUCTION. 
Preliminary Principles - : - fe eT 


ELEMENTS OF THE DIFFERENTIAL CALCULUS. 

Of Second, Third, &c. Differentials - - - - 
Of the Differentials of Sines, Cosines, &c.  - - - . 
Of Logarithmic Differentials ‘ < : 2 % 
Of the Differentials of Exponential Quantities — - . - 
Application of the preceding Rules - - - - - 
Application to the Subtangents, Tangents, Subnormals, &c. of 

curved Lines - - - - - - 
Of Multiple Points -— - - 2 ‘ ‘ 3 
Of the visible and invisible Points of Inflexion - - 
Observations on Maxima and Minima~ - - - - - 
Of Cusps of different Species, and of the different Sorts of Con- 

tact of the Branches of the same Curve - - - - 
On the Radii of Curvature and the Development or Evolute_ - 


ELEMENTS OF THE INTEGRAL CALCULUS. 

» Explanations - ~ - - - - - - - 

Of Differentials with a single Variable, which have an algebra- 
ical Integral; and first, of simple Differentials - - 


Of Complex Differentials whose Integration depends on the . 


fundamental Rule - fA x: 


Of Binomial Differentials which may be integrated algeb raically 
Application of the preceding Rules to the Quadrature of Curves 
Application to the Rectification of Curve Lines - 
Application to Curve Surfaces” - - - . - - 
Application to the Measure af Solidity  - - - e 

On the Integration of Qnantities containing Sines and Cosines 


On the Mode of Integrating by Approximation and some Uses 
of that Method - - - 2 


iv Contents. 


Uses of the preceding Approximations, in the Integration of 


Different Quantities - - - - - - - 119 
By the Table of Increasing Latitudes or Meridional Parts 132 
By Reduced Maps or Mercator’s Chart - - - - ib. 


On the Manner of reducing, when it is possible, the Integration 
of a proposed Differential, to that of a known Differential, 


and distinguishing in what Cases this may be done - 133 
On Rational Fractions - - ; - - - - eo 137 
On certain Transformations by which the Integration may be 

facilitated - - - - a - ~ 144 
On the Integration of Exponential Quantities - - - ie 
On the Integration of Quantities with two or more Variables 148 
On Differential Equations - - - - - - 151 
On Differential Equations of the second, third, and higher orders 164 

NOTES. 
1. Natureand Construction of a Curve passing through certain 

given Points - - - - - #- - - 175 
9. General Demonstration of the Binomial Formula - S77 
3. On the Method of Indeterminate Coefficients - - 179 


4, On the Methods which preceded, and in some measure 
supplied the place of the Infinitesimal Analysis. 


ist On the Method of Exhaustions - - “ - 182 
2d On the Method of Indivisibles «gis - . 185 
3d On the Method of Indeterminate Quantities - - 188 


4th Of Prime and Ultimate Ratios. - - 2 : 193 


ERRATA. 


, Page 6 line 19 for last — h? read + h? 


12 SUN see bay ar et be 
: 16 25 d.y — d.y" 
17 pA (e + 6)? — (a + a)? 
30° GBF * a2 — y2 
24 f) b—b — b—b 
d x d x dx dx 
#0 oe ten CTC Ag 
27 % number read member 


eg a Ae aes 
ame y (ET apead yy [2 Ly 
d y? dy? 
32 25 the 4 read _ the point 4 


34 13 belong to the family |» q_ § belongs to the fami- 
of the hyperboles ly of hyperbolas, 


35 11 multiples read multiple 


40 29 CAP — CPA 
45 24 and 7) — and I”, 
26 = 0 — en 
38 QR, QR — QR, QR 
49 8 AC — AC’ 
51 38 Sae — «dh 
56 8 CE — CF 


15 and other places where the word occurs, for tra- 
pezium read trapezoid 
59 22 6 by read by 6 


61 6 0 — O 
62 ia 6 bady? — 6 bada?. 
63 12 x — a 
18 Nmm_ read Nm m! 
15 8 or by means of read or values by means of 
75 33 3an° _- Zax3 
79 ti gadu (a+ bx)? read gdx (a+ bx)? 
79 oe constant read complex 
80 12 89 read 8&8 
84 10and 21 93 ao 32 


17 91 — 90) 


Vili 


88 


96 
97 
seu] 
101 


106 
118 


118 


121 


128 
131 
134 
139 
141 
152 


153 


154 


157 


158 


160 


feb | 


20 


Errata. 


30 — 29 
hyperbole — hyperbola 
pps 1 aed "ee 

St — Sst 

csse = case 
angula — ungula 

u — wu 


/1—(1i—x)? read fi —(§ — a)? . 


é 


number read member 


ie 92 » 
1.2 Birarmiee 2 
aada | aads 

aatee Kt SS. 

in — is 

give — gives 

exponents — coefficients 

+ x. — XK 

a® + 22 — a? —2?. 

number — member 

113 ae 114 

m — WU ams 7 ——— Ti 


insert a comma after the parenthesis 
/1+ 2z read “4 1—2zz 
—(y¥/ 1 read += (y¥ /—1 
—CY1 — C—YT 

— 1 — 2? read z? + 1 

a/ Ie 1p read 4/1 


gtk gtk 


INTRODUCTION. 


Tue Infinitesimal Analysis, as presented in the following 
treatise, proposes to ascertain the relation of definite, assignable 
quantities by comparing them with quantities which are here call- 
ed infinitely small. But by infinitely small quantities is meant 
quantities which may be made as small as we please, without alter- 
ang the value of those with which they are compared and whose ra- 
tio is sought. The first idea of this calculus was probably sug- 
gested by the difficulties which are often met with in endeavour- 
ing to express by equations the different conditions of a problem, 
and in resolving these equations when formed. When the exact 
solution of a problem is too difficult, it is natural to endeavour to 
approximate as nearly as possible to an accurate solution, by neg- 
lecting those quantities which embarrass the combinations, if it is 
seen that the neglect of them will not, by reason of their small 
value, materially affect the result. ‘Thus, for example, it being 
found very difficult to discover directly the properties of curves, 
mathematicians would have recourse to the expedient of consid- 
ering them as polygons of a great number of sides. For, if a 
regular polygon be inscribed in a circle, it is manifest, that these 
two figures, although they can never coincide and become the 
same, approach each other the more nearly in proportion as the 
number of the sides of the polygon increases. Whence it follows, 
that, by supposing the number of sides very great indeed, we may, 
without any very sensible error, attribute to the circle the proper- 
ties which are found to belong to the inscribed polygon. And if, 
in the course of a calculation, we should find a circumstance in 
which the process would be much simplified by neglecting one of 
these exceedingly small sides, when compared with a radius, for 
example, we might evidently do it without inconvenience, since 

i 


Fig. 1. 


2 Introduction. 


the error which would result would be so extremely small, that it 
need not be noticed. 

Let it be proposed, for example, to draw a tangent to the point 
JM of the curve 4.MB (fig. 1) considered as part of the circum- 
ference of a circle. 

Let Q be the centre, and 4PQ the axis; call the absciss 4P, 
#, and the corresponding ordinate PM, y, and Jet T’P be the sub- 
tangent sought. 

To find this, we consider the circle as a polygon of a very great 
number of sides, and im as one of these sides; we produce 
JM m until it meets the axis, and it is evidently the tangent in 
question, since it does not penetrate the polygon. We let fall up- 
on 4@ the perpendicular m p, and call the radius of the circle, a. 
The similar triangles Mrm, TPM, give 

Noriega WB cePio ee 
rm y 
Now, since the equation of the curve for the point JV is 


yy =2ar—ee, 
it will be, for the point m, 
(y+rm)? =2a («+Mr) — («+Mr)?, 


developing, we have 


SLi ts —— — 2 
y? +2y.rm+rm=2axr+2a.Mr—a2? —20.Mr—Mr, 
from which, if we subtract the equation for the point JV, we 
have 


se de es os — «2 
2y.rmt+rm=2a.Mr—22e.Mr—WMr. 
Whence, by reducing, 
ae Qy+rm 
rm 2a—2xe%—Mr 


Mr. uaa 
Substituting for — its value found above, and multiplying by y, 


we have 
__y- Qy+brm) 


~ 2a—2xex—Mr 
If now rm and Jr were known, we should have the value of 
TP sought; they are however very small, siace they are each 
less than M m, which is itself, by supposition, very small. ‘They 
are moreover perfectly arbitrary, since there is nothing in the sup- 
position to limit their magnitude, and they may be rendered indefi- 
nitely small without affecting the lines TP and PM, with which 


Introduction. 3 


they are compared. We may therefore neglect, without any ma- 
terial error, these quantities so small, compared with the quanti- 
ties 2 y and 2a—22, to which they are added; and the equa- 
y? 
a— x 
neglecting the very small quantities mr and Mr, is not only nearly 
accurate, as might be supposed, but absolutely exact, as is thus 
shown. ‘The similar Spee wo MPT, give 
QP : 5 PERO Ls 


tion is thus reduced to 7'P = 


. This value, thus found by 


whence 


PTR = = ++, asiabove. 
a—x 


The above result is thus obtained by a balance of errors; the 
error made in supposing the circumference to be a polygon being 
compensated by neglecting the small quantities mr and Jr to 
obtain the final value: and the omission of these quantities is not 
only allowable, but is absolutely necessary to fulfil the conditions 
of the problem.t 

As a second example, we suppose that it is required to find the 
surface of a given circle. 

We here also may consider the curve as a regular polygon of a 
great number of sides. The area of a regular polygon is equal 
to the product of its perimeter, by half of the perpendicular let fall 
from the centre upon one of the sides. ‘Therefore the circle, 
considered as a polygon of a great number of sides, is equal to 
the product of its circumference by half the radius ; a proposi- 
tion which is no less exact than the result found above. 

In the examples just given, it is seen that great advantage is ob- 
tained by the employment of quantities which are very small 
compared with the principal quantities in question. ‘The same 
principle once admitted may be very generally applied ; all other 
curves may, as well as the circle, be considered as polygons of a 


t If it be asked, how we may be sure in similar cases, that the com- 
pensation of errors has taken place, it may be observed, that the error, 
if any exist, depends upon the arbitrary quantities rm and Mr, and 
may be wade as small as we please by diminishing these quantities ; 
but as these disappear in the final result, the error AeARpeAtS with 
them, and leaves the result perfectly accurate. 


4 Introduction. 


great number of sides. All surfaces may be considered as divid- 
ed into a multitude of zones, all bodies into corpuscles ; in short, 
all quantities to be decomposed into small parts of the same kind 
as themselves. Hence will arise many new relations and combi- 
nations, and it is easy to judge, from the examples given above, 
of the resources furnished to the calculus by the introduction of 
these elementary quantities. 

The advantage obtained is even much greater than we should 
at first expect, since, in many cases, as we have seen above, the 
method employed is not merely an approximation, but leads to 
perfectly accurate results. It becomes therefore an interesting 
object, to ascertain when this is the case, to extend the application 
of the principle, and to reduce the methods employed to a strict 
and regular system. Such is the object of the infinitesimal 
analysis. 

We shall now give some problems, tending to throw light on the 
mode of reasoning employed in this calculus. 


1. To draw a tangent to the common cycloid. 


Let JEB (fig. 54) be a common cycloid, of which the gene- 
rating circle is E pq. The principal property of this cycloid 
is, that for any point m, the portien m p of the ordinate, compris- 
ed between the curve and the circumference of the generating 
circle, is equal to the arc Ep of that circumference. 

Draw to the point p of this circumference a tangent p T’, and 
let it be required to find the point 7’ where this tangent is inter- 
sected by m T’, the tangent of the cycloid. 

In order to this, we draw a new ordinate n q infinitely near to 
the first m p, and through m draw mr parallel to the little arc p q, 
which, as well as mn, we consider as a straight line. 

It is evident that the two triangles mnr, T'm p, will be similar, 
and we shall consequently bavemr:nr:: Tp:mp. But 
since, by the properties of the cycloid, we have Eg =n q and 
Ep=mp, we shall have, by subtracting the second of these 
equations from the other, Eg—Ep=ng—mp, or pg=nr, 
ormr-==nr. Wherefore, by reason of the proportion found above, 
we have 'p=mp, or 'p=E p, that is, the subtangent T'p is 
always equal to the corresponding are Fp. ‘This equation is dis- 


Introduction. ‘5 


engaged, by the disappearance of mr and nr, from every consid- 
eration of infinite or arbitrary ; whence the proposition is rigor- 
ously and necessarily exact. 


2. To show that in motion uniformly accelerated, the spaces de- 
scribed are as the squares of the times, reckoning from the be- 
ginning of the motion. 


In this motion, the accelerating force acts constantly in the same 
manner, wherefore, if we suppose g to be the velocity communi- 
cated in each unit of time, the successive velocities will evidently 
form the series g, 2 g,3 g, 4g, &c.; so that after a number of units 
of time marked by ¢, the velocity acquired will be as many times 
g as there are units in ¢, that is, calling the velocity u, u will 
equal gt. 

Since the velocities g, 2g, &c. are each nothing but the space 
which the moving body describes in the corresponding interval of 
time, the total space described during the time ¢ will be the sum of 
the terms of this arithmetical progression. But the sum of the terms 
of such a progression is found by multiplying the sum of the first 
and last terms by half the number of terms. Whence, this sum 
will be, (substituting w for its value gt, which is the last term) 


(g+u) x i Whence, if we represent the space by s, we have 


t 
s=(g+u)x ie: 


Let us now conceive that the accelerating force acts without 
interruption, or, which is the same, that the time is divided into 
an infinite number of infinitely small parts called instants, and, 
that at the beginning or end of each instant, the accelerating force 
gives a new impulse to the moving body. We conceive, more- 
over, that it acts by infinitely small degrees. Then g, being infi- 
nitely small compared with u, which is the velocity acquired in 
the infinite number of instants indicated by t, we must, in the 
equation s = (g + u) ey omit g, and we shall have s = s, 

If we call the velocity acquired at the end of a second, p, the 
velocity acquired after a number, ¢t, of seconds, will bet >p. 


; 5 ut : 
Whence u=pt. ‘The equation s= —, found above, will thus 


6 Introduction. 


. tt . 
become s =f. If therefore we represent by S another space 


described in the same manner during the time 7’, we shall in like 


T2 
manner have S = es , whence we may conclude 


ott ip TL , 
$0) Sie Se ee 
2 2 . 
which was to be proved; and which being freed from all consid- 
eration of infinite, is necessarily and rigorously exact. 


3. To determine in what manner to divide a quantity, a, into two 
parts, in such a manner that the product of these parts shall be 
the greatest possible. 


Let x be one of the parts, the other will be a—a, and the 
product will be ax—2?. Let this product be supposed the 
greatest possible product of the two parts of a. Suppose a to 
take a new value infinitely little differing from its present value. 
Let this be 7 -+A. This value, substituted in the above product, 
givesaxtah—a*—2xrxh—h?. If we subtract the former 
value ax —zx? froin this, there will remain ah—2hax—hA?. 
Now since ax — x? was by supposition the greatest product pos- 
sible, this increase must be nothing; therefore 

ah—2he—h? =0; orah=2ha—h?. 
But A? is infinitely small compared with 24 2, since it is an infi- 
nitely small part of an infinitely small quantity, and may there- 
fore be neglected. We therefore have ‘ ~ 
ah=2he, or @= 2a, or b=, 
on the supposition that the product is the greatest possible. 
Whence we conclude, that each of the two parts is one half of a. 

These examples are introduced to show how the principles of 
the infinitesimal analysis may be employed in ordinary reasoning 
and in common algebra. In the following treatise the same prin- 
ciples are reduced to a system in the differential and integral cal- 
culus. 


1 
1,000,000,000,000° 


tif his only Sig = 


PRINCIPLES 


or 


THE CALCULUS, 


SERVING AS 


AN INTRODUCTION 


TO THE 


PHYSICO-MATHEMATICAL SCIENCES. 


Preliminary Principles. 


Aerzra and the application of Algebra to Geometry con- 
tain the rules necessary to calculate quantities of any definite mag- 
nitude whatever. But quantities are sometimes considered as vary- 
ing in magnitude, or as having arrived ata given state of magnitude 
by different, successive variations. The consideration of these va- 
riations gives rise to another branch of analysis, which is of the 
greatest use in the physico-mathematical sciences, and especially 
in mechanics, in which we often have no other means of deter- 
mining the ratio of quantities, which enter into questions relative 
to this science, than that of considering the ratios of their varia- 
tions, that is to say, of the increments and decrements which they 
each insiant receive. | 

As an introduction, therefore, to Mechanics and the other 
branches of Natural Philosophy, it is well to obtain some knowl- 
edge of this part of the calculus, the object of which is, to de- 
compose quantities inta the elements of which they are compos- 
ed, and to ascend or go back again from the elements to the 
quantities themselves. ‘This is, strictly speaking, rather an ap- 
plication of the methods, and even a simplification of the rules of 
the former branches of analysis, than a new branch. 

2. We propose to ourselves two objects. ‘Lhe first is, to show 
how to descend from quantities to their elements ; and the meth- 
od of accomplishing this is called the Differential Calculus. The 


§ Differential Calculus. 


second is, to point out the way of ascending from the elements 
of quantities to the quantities themselves; and this method is 
called the Integral Calculus. 

So much of these two methods, as is of essential importance, 
may be easily understood, as it is but a consequence of former 
parts of analysis. Those branches of these methods which re- 
quire more delicate researches, or which are of a less frequent 
application, will be distinguished by being printed in a smaller 
type. 

3. As we are about to consider quantities with relation to their 
elements, that is to say, their infinitely small increments, it is 
necessary, before proceeding farther, to explain what is meant by 
quantities infinitely small, infinitely great, &c., and to point out 
the subordination which must be established between these quan- 
tities in calculation. 

4, We say that a quantity is infinitely great or infinitely small 
with regard to another, when it is not possible to assign any quan- 
lity sufficiently large or sufficiently small to express the ratio of 
the two, that is, the number of times that one contains the other. 

Since a quantity, as long as it is such, must always be suscep- 
tible of increase and diminution, there can be no quantity so 
small or so great, with regard to another quantity, but we may 
conceive of a third infinitely smaller or greater. For example, 
if w is infinitely great with regard to a, although it be then impos- 
sible to assign their ratio, this does not prevent our conceiving of 
a third quantity, which shall be to a, as @ is to a, that is, which 
shall be the fourth term of a proportion, of which the three first are 


Phy ckhaee 
a:«e:: a: 3; this fourth term, which is —, must therefore be 
a 


infinitely greater than a, since it contains 7 as many times as & 
is supposed to contain a. In the same manner, nothing prevents 
our conceiving of the fourth term of this proportion, v:a::a: 23 


atl te 7) Lie Hiakeok 
and this fourth term, which is —, will be infinitely smaller than a, 
xv 


since it 1s contained by a as many times as a is supposed to be 
contained by x ‘There are no bounds to the imagination in this 
respect; and we may still conceive of a new quantity, which 


oes k pd See 
shall be infinitely smaller with regard to iis than — is with regard 


Preliminary Principles. 9 


toa. We call these quantities infinitely great or infinitely small 
quantities of different orders. 

In general, the product of two infinitely peat quantities or of two 
infinitely small quantities of the first order, is infinitely greater or in- 
finitely smaller than either of the two factors ; for, my: yi:a:1; 
but, if w is infinite, it contains unity an infinite number of times, 
a y, therefore, contains y an infinite number of times. A similar 
course of reasoning shows that a product or a power, of any num- 
ber of dimensions whatever, and all of whose factors are infinite, 
is of an order of infinite marked by the number of its factors ; 
thus, when @ is infinite, v* is an infinite of the fourth order, that 
is, infinitely greater than 2°, which is infinitely greater than 2, 
which is itself infinitely greater than a. For 

BOE Seow es Cet Shean, ret date eh itase @ dak Sy V's 
On the contrary, if x were infinitely small, «+ would be then an 
infinitely small quantity of the fourth order, that is, infinitely 
smaller than #%, while vw would be infinitely smaller than 2, 
which would be infinitely smaller than 2. 

Again, a fraction whose numerator is a finite quantity, and 
whose denominator is any power of an infinite quantity, would be 
of an order of infinitely small quantities, marked by the exponent 


b : ; j 
of that power. ‘Thus, ory for example, is an infinitely small 


. . e ° . b ° ° 
quantity of the second order, if @ is infinite ; and - , an infinite- 
xv 


ty small quantity of the third order. For 


Lie Hs: teistlia. 
Cede 8 Bea ag 
But if a product have not all its factors infinite, then its order 
of infinity is to be determined by the number of those factors 
only which are infinite; thus aw y, for example, is of the same 


order asvy; since avy: ay ::@: 1, and this last ratio, 
a1 OF = isa determinate ratio, if a is a finite quantity. 


This difference is worthy of observation in the comparison of 
infinitely great or infinitely small quantities with each other or 
with other quanties, with regard to which they are infinitely great 
or small. If be infinite with regard to a, nothing can measure 
their ratio; but, on the same supposition, the ratio of x, to 2 mul- 

2 


10 Differential Calculus. 


tiplied or divided by any finite number whatever, is a definite ra- 
tio. For example, x being infinite or infinitely small, is not com- 
parable to a, a being supposed a finite number; but it may be 
compared with aa, sincex« : axr::1: a. 

5. ‘l'o express, in calculation, that a quantity 2 is infinite with 
regard to another quantity a; or, which is the same thing, to ex- 
press that a is infinitely small with regard to a, we must, in the 
algebraical expression where these quantities are found together, 
reject all the powers of « lower than the highest, and consequent- 


ly all those terms without «. If, for example, in 2 is sup- 
posed infinite with regard to a and 6, we must suppress a and 6, 
ke 3 3x-+a4 - aie 
and we shall have ole for the value of 5 pad when 2 is in- 
S04 Bias 
finite. For —* is the same thing as dividing the nu- 
2 


merator and denominator by #; and, when we suppose 2 to be 
itt ; a b ; 
infinite with regard to a and 6, the fractions ve and me) which re- 


present the ratios of a and 6 to x, must necessarily be suppressed, 
since, by this supposition, these ratios are less than any quantity 
whatever; wherefore, in this case, the proposed quantity is re- 


3 
duced to = 


The quantity «? +a«-+ would, in like manner, be reduced 
to x2, on the supposition that w were infinite. For it is only on 
ithe supposition that 6 adds nothing to the value of az + 6 that x 
can be said to be infinite: and, in the same manner, it is only by 
supposing that av adds nothing to the value of x? +a that 2? 
can be said to be infinite. Wherefore, both az and 6 must be 
considered as of no value by the side of x?, and are therefore to 
be rejected, and the quantity is reduced to x?. 

If, on the contrary, 2 were infinitely small, it would be neces- 
sary to retain those terms only in which the exponent of @ is 
smallest. Thus v? +a is reduced to aa, when z is infinitely 


axc+tb. : b 
and Beirey in the same case, is reduced to —. s 
cu+td 


small ; 7 


Preliminary Principles. epee 8 


{t need not be apprehended that these omissions will affect 
the consequences to be drawn from the calculations in which they 
may be made. On the contrary, it is only by these omissions 
that we can express what we mean to express, viz. that 2 is infi- 
nitely great or infinitely small. It is only by these omissions that 
we can arrive at a conclusion conformable to the supposition 
which we have made. [or if, when we supposed @ infinite, we 
‘should not reject the terms just pointed out; if, for example, in 


Be 
Seta, we een es Na 
or ——-, we should not reject — and —, then the cal- 
b ; 
5xex+6 pel 4 x 
a 


a b 
culus, not expressing that — and = are ratios less than any as- 
ae 


signable quantity, would not answer to what it is required to know, 
viz. what is the value of that quantity when w is infinite ; in short, 


by allowing © and ~ to have any effect on the value sought, we 
2 x 


contradict the supposition which we have made, that 2 is infinite 
We shall not want occasions for verifying the exactness of this 
principle of neglecting infinite quantities of the inferior orders. 
For the present, the following example will confirm the reasoning 
we have just made us of. Let there be the series 4, 2, $, 4, 5, §, 
&c. ; the terms of this series evidently approach nearer and near- 
er to unity, yet without ever being able to pass this limit. Now 
x 
wxtl 
number expressing the place of that term. Since then the terms 
continually approach unity, and that the more nearly, as they are 
farther from the beginning of the series, they can reach that limit 
only at an infinite distance from the beginning of the series; in 
order, therefore, to express the last term of this series, we must 


each term may be represented by 


, by substituting for a the 


suppose in that w is infinite; but, conformably to the prin- 


x+il 


3 ° ° x . 
ciple, this quantity must then be reduced to —, that is to say, to 1; 
P ,] q y x 7 ? 


veh . ° x 

the omission, therefore, of the term + 1 in the expression eae 
so far from making the conclusion false, is, on the contrary, that 
which makes it what it ought to be. In short, by making this 


12 Differential Calculus. 


omission, we act conformably to the supposition which has been 
made. 

Such is the subordination which must be established in the cal- 
culus, between infinitely great and infinitely small quantities of 
different orders. But in the application of this principle of the 
omission of quantities, certain particular cases may occur which 
it will be well to notice. 

Suppose we have the two quantities 

x? Sax 6)vand: @ Paw ex 
when «@ is infinite, each of these is evidently reduced to x?, so 
that their difference, in this case, seems to be nothing. But if we 
take this difference according to the common rules, we find it 
b —c, or c—O, whether z be infinite or not. This seeming dif- 
ficulty, however, is easily solved. For the difference of these 
quantities is really 6— c or e—63 but when we seek this differ- 
ence, after having supposed @ infinite in each, it is the same as 
asking what this difference is, compared to the quantities them- 
selves; and, since each of them is infinite, we ought to find, as 
we do in fact, that the difference is nothing in comparison with 
them. When, therefore, it is asked what the result of certain 
operations on several quantities becomes, on the supposition that 
x is infinite, it is to the result that we must apply the principle 
stated above, and not to each of the quantities taken separately. 
Thus we shall find that the sum of 
—zr* +ar+6, and c?+6b7-+4¢, 

is reduced, when x is infinite, toaxr-+6; for, by the general 
rule, itisaxv+6a¢+6-+¢, which, when a is infinite, is reduced 
toax+b«ax. In like manner, if we had 7 —4/z2—2:2 ; this 
quantity, when a is infinite, seems to be nothing. But as 4/22 G2 
is only an indication of the root of 2? — 67, we must, in order to 
find the difference between this quantity and w, reduce x2 — 62 
to a series (ig. 144); the quantity 7—4/22—52 will then 


b2 b+ 52 b+ 
become ~— 27 a oe + Rips tes OF aes 


+ &c., which. 


Bat 8 x3 


bebe aA p AN bs ; . 62 
when vz is infinite with regard to 5, is reduced to ae 


Simple Quantities 13 


Elements of the Differential Calculus. 


6. When we consider a variable quantity as increasing by infi- 
nitely small degrees, if we wish to know the value of those incre- 
ments, the mode which most naturally presents itself is, to deter- 
mine the value of this quantity for any one instant, and the value of 
the same quantity for the instant immediately following. The 
difference of these two values is the increment or decrement by 
which this quantity has been increased or diminished. ‘This dif- 
ference is also called the differential of the quantity. 

7. To mark the differential of a simple variable quantity, as x 
or y, we write dx or dy; that is, we place before the variable the 
initial d of the word differential. But when we wish to indicate 
the differential of a compound quantity, as v*, 52243 a2, or 
/xz —a2, &c., we enclose this quantity in a parenthesis, before 
which we write the letter d; thus we write 

d (w?), d(5x°+434?), d(4/z2 —a2), &c. 
The differential of a compound quantity is also sometimes ex- 
pressed by a point between d and the quantity, asd.a?,d.xyz, &c. 

We shall hereafter represent the variable quantities by the last 
letters of the alphabet, ¢, u, 2, y, 2; and the constant quantities, 
or those which always preserve the same value, by the first let- 
ters, a, b, c, &c., and if they are used otherwise, notice of it will 
be given. As to the letter d, it will be used only to designate the 
differential of the quantity before which it is placed. 

8. Agreeably to the idea which has just been given of the dif- 
ferential of a quantity, we see, that to get the differential of a 
quantity which contains only variables of the first degree, and 
neither multiplied nor divided by each other, we have only to 
write the characteristic, d, before each variable, leaving the sign 
of each unchanged for example, the differential of «+ y—2z 
will be dw + dy—dz. For, in order to obtain this differential, 
we must consider x as becoming 7+ dx; y as becoming y+ d y; 
and z as becoming z +d 2; then the quantity proposed, which is 
x + y— z, would become « 4+-dx +y+dy— z—dz; and, tak- 
ing the difference of these two states, we shall have 

rt+de+ty+dy—z—dz—ar—y+z2; 
that is, 
d(w+ty+tz)=dx+dy—dz. 


14 Differential Calculus. 


The case would be the same, if the variables, which enter into 
the proposed quantity, had constant coefficients ; thus the differen- 
tialof52+3y, is5dx+3dy; that ofar+by,isadxz+bdy; 
for, when «x and y become x+da andy+dy, the quantity 
axz+by becomes a (x1+dx) +6 (y +d y), that is 

axtadetby+bdy; 
the difference of the two states, or the differential, isadx +bdy; 
that is, generally, each variable must be preceded by the character- 
astic d. 

If in the proposed quantity there be one term entirely constant, 
the differential will be the same as if there were no such term. 
That is, the differential of that term will be nothing. This is 
evident, since the differental being nothing else but the increment, 
a constant quantity cannot have a differential without ceasing to 
be constant; thus the differential of ax +6 is simply ada. 

9. When the variable quantities are simple but multiplied to- 
gether we must observe the following rule. Find the differential 
of each variable quantity successively, as of all the rest were a con- 
stant coefficrent. 

For example, to find the differential of 2 y, we first consider « 
as constant and obtain wd yj, we then consider y as constant and 
have y da, so that, d(wy) =ady+ydza. 

The reason of thisrule will be seen by going back to the prin- 
ciple upon which it is founded. To find the differential of «xy, 
we must consider x as becoming x +d 2, that is, as increasing by 
the infinitely small quantity dw; and y as becoming y +d y, that 
is, increasing by the infinitely small quantity dy; then a y be- 
comes (x +d) x (y+dy), thatis,rytady+yde+dydz; 
then the difference of the two states, or the differential, is 

vyptedy+ydx+tdydx—xy, 

or rxdy+ydxe+dydz; 

but in order that the calculus may indicate that d y and dx are 
infinitely small quantities, as they are supposed to be, we must (5) © 
omit d yd a, which (4) is an infinitely small quantity of the sec- 
ond order, and therefore infinitely small compared with wd y and 
y d x, which are infinitely small of the first; therefore the differ- 
ential of eyord.xyisrdy+y da, agreeably to the rule. 


+ To avoid obscurity, in the manner of writing, it is best to write 
last the variable affected with the characteristic @. 


Compound Quantities. 15 


We shall find, by the same rule, that the differential of x yz 
iseydz+azdy+yzda, by differentiating as if ry, xz, and 
yz were successively constant quantities. ‘l'his may be demon- 
strated, as above, by regarding a, y, and z as becoming, respec- 
tively, c+da,y+dy, and z+dz; in which case vayz is 
changed into (ew +dza2) (y+dy)(z+dz)=axyz4+nydz+ 
vedytyzdr+ydadz+zdydzr+adzdy+dardydz; 
then the difference of the two states, will become, by reducing, 
and rejecting the infinitely small quantities of the second and third 
orders, 

d.vyz=uydz+uzdy+yzdua. 

10. If the quantity proposed be any power of a variable quan- 
tity, observe the following rule. Multiply by the exponent, di- 
minish this exponent by unity, and multiply the result by the differ- 
ential of the variable. 

Thus, to find the differential of #?, we first multiply by the ex- 
ponent 2, diminish this exponent 2 by 1, then multiplying by da 
the differential of the variable «2, we have 2xdx. 

We shall find, in the same manner, that the differential x3 is 
32% da; that of x*,4a% dix; that of a7!,— a* da; that of 


as,—3a*d ax; thatof x?, a a? da ; that of as, is 4 atdas 
and, generally, that of a”, is ma™~'d «3; whether the exponent 
m be positive or negative, a whole number or a fraction. 

To find the reason of this rule let us go back to the first prin- 
ciples. Let us consider x as becoming «+d & (d x being infi- 
nitely small) ; then a” becomes (a + d x)”, which, being reduc- 
ed to a series (Allg. 144) becomes 


™ 
x me Be od 2 im 


= Lem 2 dx? 4 &e. 
or, because the term m = .x™—2d x% is infinitely small of 
the second order, and the following terms would be of still lower 
orders, the series is reduced to a” 4+ma™—1d «x; then the differ- 
ence of the two states is 2” +ma™—1da#—a™ ; and, there- 
fore, d.x™ =ma™—1"d x. 

if there were a constant coefficient, the case would not be al- 
tered, the constant coefficient would remain in the differential the 
same that it is in the quantity ; 

,d.ax™ , therefore, = maa™7da. 


16 Differential Calculus. 


We have thus given whatever it is necessary to know, in order 
to be able to differentiate all sorts of algebraical quantities. What 
follows is only an application of these rules. 


11. Suppose it were required to differentiate the fraction ~, we 


should write it x y~* (Alg. 133); and then, applying the rule 
given (9), we have 

ad.2y-*=ard.y-*+y-1d a, 
and, consequently, (10)d.xy-t=y-'dxax—ay-*dy =, by 
EEE aaa 


“ 


reducing to a common denominator, 


Therefore, to find the differential of a fraction, we multiply the 
differential of the numerator by the denominator, subtract from the 
product the differential of the denominator multiplied by the nu- 
merator, and divide the whole by the square of the denominator. 
This is the rule usually given for the differentiation of fractions ; 
but we easily perceive that we may dispense with charging the 
memory with this new rule; as itis sufficient to raise the denom- 
inator into the numerator (lg. 133), and*then differentiate by 
the general rule. 

12. If we wish to differentiate aa* y?, we first consider «* 
and y? as two simple variables, and (9) we have | 

dtaxsy* =ax* doy*+ ay? doa; 
then (10) we have 
d.ax?yt*=2Qaxi ydy4d3ay? x dx. 

In general, 
d.ax™y" =ax™d.y +ay*d.a™=nax™y"—dytmay®a™—ldax. 

13. If the quantity, which we wish to differentiate, is complex, 
but without containing any powers of complex quantities, we 
differentiate separately each of the terms of which it is compos- 
ed. Thus, 

d (ax +bx?4-cxry) = 3axn2*dx4+2bada+cudy+cydx. 


In hike manner, 
d (ax? +b 40 =d(ax? +b x+4c4-*y) 


=2axdx+bdx—2cx*ydx+cu*dy. 
In like manner, 
d(aty+ay?+b5)=3a7% ydxe+ardy+2aydy, 
observing that the constant 6% has no differential. 


Complex Quantities. 17 
14. If the exponent of the quantity be a whole number, as in 
(a+ba+ca*)>, we regard the whole quantity affected by this 
exponent as a single variable, and differentiate by the rule for 
powers (10). Thus, 
d(a+bxax+cu?)' =5(a+ba+ceu?)*xd(at+bu+ex?) 
=5(a+bx+¢2%7)* x (bdxe+2cxdx). 
In like manner, 
d (a +bx2)3 =4(a+ba2)= xd (a+b x2) 
= (a4+bu2)8 x Qhadx='tbadau(atb«*)s. 
d(x* +2ax+a?)? =4 (x? + Diirke 143 (~-+a)d x. 
15. When a complex quantity is composed of different factors, 


we regard each factor as a simple variable, and follow the rule 
given (9) for a product of several simple variables. Thus 


5 
x3 (a + 6 x*)5, which we may consider as composed of the two 
7 5 . . 
factors a? and (a + 6 x?)5, will give 


d (a3 (a+b «2)3) =(a+b a2) d(x5)+u3d(a+b a2)8, 
which, by the preceding rules, becomes 
Za% dx(a+b a2) 449 bxtdax(a+bx2)s. 
And 
ae )=e ; (x -- a) (a + b)-?} — (~-+a)8 d (2 4b)-? 
(a + 6)? 
+(x +5)-2d (a 48)8; 
that is, 
=—2(x+a)3(x+b)--da+4+3 (x 4+b)-? (x +a)? dx, 
which, by restoring the denominators, becomes 
__ 2(w+asda , 3(a+a)Prdax _ 
“ery + ery > 
(reducing to a common denominator) 
2(a + a)* (w+a)da ae 3 (x + a)? (w +b)dx 
(a + b)8 (xv + b)3 
_ —2(a@ +a)? (w+ a)dx +3 (e+ a)? (w +b) dx 
4s (Te 
(3.x+3 b—2x—2a)(x-+a)? dx _ (x+3 b—2a) (xw+a)2?dax 
ae i (w +5)5 


a(nda-+yd 
Also,  comenre i)=-" een Lani 
(0? < a2)? (a? + 2)? 


{8 Differential Calculus. 


16. If the proposed quantity is radical, we substitute fractional 
exponents in place of the radical signs (4/g. 132), and differ- 
entiate according to the rules already given. ‘Thus, 


iu 1 a 
2) =} i 
d (Vz )= d (x3) = x Fda oe a 
5 3 es) 
d (yas) =d(a?)=3% Fdx; 
d (Vat 73) =d (a? a)? = 3 (a? — a?) ? d (a? — 29) = 
~1 —w2«d 
re oem ‘hare 
( q p Bu: 
dam Vapomy §=d lam (apdar)e } 
e. Pits 
=a" d(a+ba")? 4 (a+b eee AS es 


=n ibe amtr-ldax (a+bar)i~ $m x™—3 da (a +b 2") 


In like manner, 


£ 
q 


1 
2 


d (x+y)? = ia py Fd(w+y? ‘)= pyidethdy 


(acy + 2) 
(as — Sade U)=-oS 
(a—.x)* (a—x)* L+ta/J (1+)? 
d an Stee he el Oe 


Of Second, Third, &c. Differentials. 


17. In addition to the differentials which we have just been con- 
sidering, and which are called first differentials, we consider also 
second, third, &c. differentials. These are indicated by writing 
twice the characteristic, d, before the variable, for the second differ- 
entials, three tunes for the third, &c. For example, dda indicates 
the second differential of 7; ddd, the third differential. 

When we speak of second differentials, we consider the variable as 
increasing by increments which are unequal, but whose differential is 
infinitely small with regard to these increments themselves. Thus 
dd x is infinitely small compared with dz. In the third differentials 
also, ddd x or d? & (for they are indicated in both ways) is infinitely 
small compared with dd x, and so on. To indicate the square of d a, 


+ From this expression we may deduce the rule, to differentiate a radical of the 


scond degree, we divide the differential of the quantiiy under the radical sign, by 
double the radical self. 


Successive Differentiations. 19 


we should naturally write (d a)? ; but, for greater simplicity, we write 
d x2, which cannot be mistaken for the differential of a2, as that is 
designated by d (a?) or d. x?, 

We observe that although dd x and d x? are both infinitely small 
of the second order, they are nevertheless not equal ; for dd x is the 
second differential of 2, or the difference of two successive differen- 
tials of a; and d x? is the square of d x. 

In jude to determine the second differentials, it is nist natural to 
consider the variable quantity in three successive ‘states, infinitely 
near each other; to take the difference between the second state and 
the first, that between the third and second, and then take the differ- 
ence of these two differences. For example, the first state of ais x ; 
at the second instant it has increased by the quantity da, and become 
x+dx; the following instant « + dx increases by d x +d (da), 
d (dx) marking the quantity by which the increment of the second 
instant exceeds that of the first, or the differential of da. Thus the 
three successive states of the quantity x are 

v,u+dxx2x+2da+d(dea). 
The difference between the second and first is dv; that between the 
third and secondisdx-+d(dax); finally, the difference between 
these two differentials, or the second differential of a, is d (dx); we 
have therefore dda =d(dax). Therefore, the second differentials 
are obtained by differentiating the first differentials according to 
the rules already given. 

For example, to get the second differential of x y, we take the first 
differential, which is «dy + ydx3; we then differentiate this quan- 
tity asif wand dx, y and dy, were so many different variables, and 
we find wddytdydv+yddx+dyda; 
or dd.xcy=uddy+2dydx+yddax, 

In like manner the second differential of «2 is found by first dif- 
ferentiating x?, which gives 27d; then differentiating 21d as 
if aand d x were both finite variables, which gives 

Qxrddx+2d x2. ' 
We shall find also, that 
dd.ax™=d.maa™d2 =m.m—tax™ 2d x2+max™-1d da.t 

We should proceed in the same way to differentiate a quantity in 
which, there were already first differentials, whether it were the re- 
sult of an exact differentiation or not. ‘Thus 


t A difficulty may present itself in this mode of taking the second differentials 
which it is best to explain. When we determine the first differentials, we reject 
the infinitely small quantities of the second order; but the second differentials 
being infinitely small of the second order, is there not reason to fear that what 


20 Differential Calculus. 


d.xcdy=xuddy+dady. 


a. ot .eldya=—an*tdady +a 'ddy. 
dx ny i, tS ddx dxddy 


18. It often happens that in calculations into which several varia- 
bles enter, the first differential of one of the variables is supposed. 
constant. This supposition is allowable, because we may always 
take one of the first differentials as a fixed term of comparison for the 
other first differentials, and it simplifies the calculation, inasmuch as 
the terms affected by the second differentials of this variable will not 
afterwards occur in the course of the operation, since if dais con- 
stant, dd « = 0, which makes all the terms affected by d da disap- 
pear. We have only to take care, in this case, not to differentiate 
dx (or the constant differential) in the terms in which it occurs. 


Thus, the differential of - or d.dxdy~? is, on the supposition 


dxdd 
that d xis constant, —d xdy-?ddy or — Tape If, on the con- 
d Vnitr ees 
trary, we suppose @ y constant, 1t 1 lap 


19. By reasoning in the same way, we perceive that the third dif- 
ferentials are found by differentiating, in the usual way, the second 
differentials, considering the variables and their first and second diffe- 
rentials as so many different variables ; and so for higher differentials. 
It is only necessary to observe, that when a differential has been con- 
sidered as constant in the passage from the first to the second differ- 
entiation, it must be considered so in all the successive differentiations. 


Remarks. 


20. We have supposed, in the preceding pages, that the varia- 
bles x, y, &c. all increased at the same time ; that is to say, that # 
becoming v +d, y became y +dy, and so of the rest. But it 
may happen that some diminish while others increase. In this 
case we must, after the differentiation, change in the result the sign 
of the differential of the variable which has diminished. Ory, in- 


we have rejected in the valuation of the first, will render the second defective ? 
We answer, no: for this infinitely small quantity of the second order, which has 
been rejected, can have for its differential only an infinitely small quantity of the 
third, which would be rejected by the side of the second differential, since a 
second differential is infinitely small of the second order. 


Differentials of Sines, Cosines, &c. 21 


deed, we may let the differential remain as given by the preced- 
ing rules, but, in the application which we make of it to the ques- 
tion, observe to take negatively the quantity which represents the 
differential of the variable which has diminished. For if y has 
been diminished by a quantity, g, and if, in the differentiation, 
you have tacitly supposed y to become y+dy, it must follow 
that y—q=y+dy, or that —q=dy, or g=—dy; in such 
cases, therefore, what has been called dy must be called —d y 
every where else but in the differentiation. We shall see exam- 


ples of this hereafter. 


It will be the same with second differentials compared with first. 
If the first differential diminishes, you will nevertheless differentiate 
in the usual way, but in the application to any question, you will, if 
dy, for instance, is the differential in question, call that —dd y, 
which you would otherwise have called d d y. 


Such are the rules for differentiating quantities, when they are 
presented directly. But it often happens that it is not so much 
upon quantities themselves, as upon certain expressions of those 
quantities that we have to operate. Instead of angles, for exam- 
ple, we often employ their sines, tangents, &c.; often, also, we 
meet with the logarithms of quantities instead of the quantities 
themselves. We proceed to show how to differentiate these kinds 
of expressions. 


Of the Differentials of Sines, Cosines, §c. 


21. If we have to differentiate such a quantity as sin z (or the 
sine of the angle or arc ~), we must conceive that the angle z 
becomes z + dz, and then sin (z + dz) — sin z is the differential 
of sinz. Now according to what has been laid down (Trig. 11), 
sin (e-+dz) =sin z cos dz+sin dz cos z, supposing the radius 
= 1. But the sine of an infinitely small arc is the arc itself, and 
its cosine does not differ from the radius; we have therefore 

sind z=dz,and cosdz=1; 
then sin (2 -+dz) =sin z 4+ dz cos z; 
therefore, 
sin (z +dz)—sinz = d(sinz)=dzcosz; 
that is to say, the differential of the sine of an arc or angle, whose 


22 Differential Calculus. 


radius is unity, 1s found by multiplying the differential of the ai- 
gle by the cosine of the same angle. 

22. In like manner, the differential of cos z, or 

cos (z +d z) — cos z= cos z cosdz — sin z sind z — cos z, 
since (‘T'rig. 11) 

cos (z+dz)=cosz cosdz—sinzsindz; 
therefore, as 
sindz=dz,andcosdz=1, 
we obtain 
d (cos z) =cosz—dzsinz— cosz=—dzsin 2; 
that is to say, The differential of the cosine of an angle, whose ra- 
dius 1s 1, 1s found by multiplying the differential of the angle tak- 
en with the contrary sign, by the sine of the same angle. 
Thus, to recapitulate, we have 
d (sinz) =dzcosz; d(cosz)=—dzsinz. 

By means of these two principles, we may differentiate any 
quantity composed of sines and cosines, without any other rules 
than those already given. 

Thus, to differentiate cos 3 z, we have 

d(cos3 z)=—3dzsin8z. 
Universally, if m is a constant quantity, 
d (cos m z) =—mdz sin mz; d(sinm2z)=mdzcosmz. 
In like manner 
d (sin z cos t) = cos t d (sin z) + sin z d (cost) 
=dzcostcosz—dtsin zsint. 
And 


d i eV cen (sin z)"—) d(sinz) = mdz cos z (sinz)"—?. 


sin z fe 
. If we had — a) which is the expression for the tangent of. 


an oe when sand = 1, since (Lrig. 8) 
COSta ed's ec SIpEMe ce ten te g 
we-should have 


sin z os 
d ( =d.sinz(cosz)~'=dzcosz(cosz)— ‘4d zsinz4cosz—? 


Cos z 
__dzcosz dz sin z* dzcos =? + dzsin z? dz 
Cos x cosz2 cos z? ~~ cos 22’ 


because (Trig. 10) cos z? +sinz*=1. Therefore, 

The differential of the tangent of an angle, whose radius is 1, is 
equal to the differential of the angle, divided by the square of the 
cosine of the same angle. 


Logarithmic Differentials. 23 


Whence we may also conclude, that the differential of an angle 
s equal to the differential of the tangent of that angle, multiplied 
by the square of its cosine ; for, since 


sin z 
d(==)= paki ae 


we have dz=cos 27 d tang z. 


. . ° . COS & . ° 
24, If it were required to’ differentiate mae) which is the ex- 


pression for the cotangent of the angle z, we should have 


cos z : : : \ 
d, —— =d.cosz sin 2~!=—dz sin z sin z~!—dzcos z? sin 27? 
sin z 
dzsin z dzcosz?  —dzsinz? —dzcosz? — dz 
sin z sinz? sin 2 ~ sin z2° 


Therefore, The differential of the cotangent of an angle, is equal 
to the differential of the angle, taken negatively, divided by the 
square of the sine of the same angle. 'The use of these differen- 
tiations will be exemplified hereafter. 


Of Logarithmic Differentials. 


25. According to the description already given (lg. 238) 
logarithms are a series of numbers in any arithmetical progres- 
sion, answering, term by term, to a series of numbers in any 
geometrical progression. 

This being laid down, let y and y' be two consecutive terms of 
a geometrical progression, of which ris the ratio, and @ and a’ 
the two first terms. Let, also, x and 2’ be two consecutive terms 
of an arithmetical progression, of which 6 and 0’ are the two first 
terms. Let us suppose, moreover, that v and 2’ are in the same 
place in the arithmetical progression that y and y’ are in the geo- 
metrical progression; in which case « and 2’ are the logarithms 
of y and y’. 

By the nature of geometrical progression (.4/g. 231) we have 
y' =ry,and @ =ra; substituting in the first of these equations 
the value of 7 deduced from the second, we have 


a’ y! a’ 
fae Toupee Sle 
a y a 


Let us now suppose that the difference between y' and yis 2, or 


24 Differential Calculus. 


that y’=y-+23 we shall have 27 * = or 1 + 77 Hs and conse- 
ae 
quently, <=f 1555 or og! —~ a 


Again, the nature of arithmetical progression gives (lg. 228) 
x —x= b’—b. 

In order to find then the relation of these two progressions, let 
us suppose that the difference a’ —@ of the two first terms of the 
geometrical progression, is to the difference b — 6 of the two first 
terms of the arithmetical progression as unity is to any number 
ms that is to say, that a’ —a : b’—b :: 1: m, we shall have 
m (a’ —a) = b’—6; substituting then, in this last equation, in- 
stead of a’ — a and b’ — 4, the values which have just been found, 


M AZ ; : ; 
we shall have = v’ — x, an equation which expresses gener- 


ally the relation of any geometrical progression to any corres- 
ponding arithmetical progression. 

Let us imagine that in each of these progressions, the consec- 
utive terms are infinitely near each other; then z, which marks 
the difference of y' and y, will be dy; and a’— a, which marks 
the difference of 2’ and x, will be dx; whence, the equation will 


mady 4g. 
Y 


be changed into 


With regard to m, which indicates the relation of the difference 
of the first two terms of the arithmetical progression, to the dif- 
ference of the first two terms of the geometrical progression, it 
will nevertheless be a finite number, although these two differen- 
ces be infinitely small, because we easily conceive that one of 
two infinitely small quantities may contain the other as many 
times as one of two finite quantities can contain the other. 


The equation meee = d «x shows, therefore, that dx, the dif- 


ferential of the logarithm of a number represented by y, is 
equal to dy, the differential of that number, divided by the same 
number y, and multiplied by the first term a of the fundamental 
geometrical progression, and by the number m, which represents 
the relation of the difference of the first two terms of the arith- 
metical progression to the difference of the first two terms of the 
geometrical progression. As this number m determines, in some 
measure, the relation of the two progressions, it is called the 
modulus. 


Logarithmic Differentials. 25 


We see then, that according to the value which m and the first 
term a of the geometrical progression are supposed to have, the 
same number y may have different logarithms. But of all these 
different systems of logarithms, the most convenient in algebraic- 
al calculations, is that in which the first term of the geometrical 
progression is 1, and in which the modulus is 1. In that case, 


the equation aed = de, which comprehends all the different 


systems of logarithms, becomes 
Oy see ign 

26. In the system of logarithms, therefore, used in algebraical 
calculations, the differential dx of the logarithm x of any number 
y, ts equal to dy, the differential of that number, divided by the 
number itself. 'This is the principle by which we may easily find 
the differential of the logarithm of any algebraical quantity. But 
before making use of it, we must observe, 

Ist. That the logarithms here spoken of are not those of the 
tables ; but it is easy to deduce the one from the other, as will be 
seen hereafter. 

2d. That since the first term 6 of the arithmetical progression 


: : : d ‘ é 
is not found in the equation ped aa Sr dx, this equation, as well as 
y 


the particular equation “! = d «© just deduced from it, are always 


true, whatever may be the first term 0, that is to say, the loga- 
rithm of the first term @ of the geometrical progression. We are 
therefore at liberty to suppose, for the sake of greater simplicity, 
that the first term of the arithmetical progression is nothing ; and, 
as the geometrical progression which has been fixed upon has uni- 
ty for its first term, we shall take zero or 0 for the logarithm of 1 ; 
but it should be observed that this is entirely arbitrary. 

By thus taking unity for the first term of the geometrical pro- 
gression, and zero for the first term of the arithmetical progres- 
sion, or for the logarithm of unity, the rules already given (4/g. 
241) for the application of logarithms will equally well apply 
here. If we generalize these rules, designating logarithm by J, 
we shall see that, instead of J (ab) we may take a+); in- 

4 


26 Differential Calculus. 


> & 
stead of lore la—tlb. Inthe same manner, /a"=mla; final- 


ly, Pm an = ~ La. 

This being laid eve if we apply the principle which has just 
been established concerning the differential of the logarithm of a 
number, we shall find that 


eee _a(ta+a) dx. 
Oe te Panty 29 peerage 
ty d(a+ x) _ d x 


observing that the differential of the constant / a = 0. 
We have also, 


d.tiad(l1—la)=—; dla? ad. 22a 2, 
Ale) Sale lee aie 

‘= ues, ‘ah 
Aha he aa ae : ak 


ai(F+*) =4(i (4 +2) -l@—2)) = de da 


atv a—x 


* 2 
dl (a? na pial +x?) Q2xrdax . 


ee ey aa ee xd x xdexv 
bb at £23 Se iE OS 


4/ a +4. x2 “/ a2 + x2 a/a2 + x2 a2 ++ 2? 2 
or, more directly, 


dlyatpr2 =d.3l (a? +a?) = ee 
di(a™(a +ban yHa(lam+l (at ba") ) 


ndx  npbx®-1dee 
= © ny) nnn 
(mlx+pl(a+ba") ) Fate He 
These examples are sufficient to show how other logarithmic 
quantities may be differentiated. 


Exponential Quantities. 27 


Of the Differentials of Exponential Quantities 


27. We sometimes meet with quantities of this form, c*, ay ; that 
is to say, quantities whose exponent is variable. They are called 
Exponential Quantities 

In order to find how to differentiate these quantities, let us suppose 


LY == Z5 
then, taking the logarithms of each number, we have 
lay =Iz 
dix. 
and consequently dl(xy) = ; 
then dz, == ebb (xl), 


and, substituting for z and d z, their values, 

d (xv) = xvdl (x4); 
that is to say, the differential of an exponential quantity is found by 
multiplying that exponential quantity by the differential of its loga- 
rithm. Thus, 


d.x¥ = ad. lav=axd.ylxa=av(dylat ay 


In like manner, 


d (av +y*)=d.av+d.y* =ard.lax + yrd.ly* 


=artd.vla-yzd.zly =ardxla+ y* (dzly + =a " 
So 
l(a? + 22)" = (a2 02)" d l(a? pat)? = (a208)"d.a l(a? 4.2%) 
Qxurtdax 
== (a? + x2)” (dal (a? + a?) + oa) ; 

and so of others. | 

Frequent use is made, in calculations, of the exponential quantity 
e*, e being the number whose logarithm is = 1. The differential of 
this quantity ts, according to what has just been laid down, 

evd .bev*=netd(xle) = erdale; 
and since leis supposed = 1, we have simply 
d.e%¥ = erdax. 

That is to say, this particular exponential has, for tts differential, the 
exponential itself multiplied by the differential of its exponent. ‘This 
exponential will be found hereafter. 


Fig. 1, 


38 Differential Calculus. 


Application of the preceding rules. 


28. In order to show, by some examples, the use of the rules 
which have just been given, and the advantage which they have 
over common algebra, we shall now apply them to subjects with 


which we are already acquainted, viz. questions in Geometry and 
Algebra. 


Application to the Subtangents, Tangents, Subnormals, &c. of 
Curved Lanes. 


29. To draw a tangent to any curve line 4M (fig. 1), we con- 
ceive this curve to be a polygon of an infinite number of infinite- 
ly small sides. ‘The prolongation MT’ of Mm, one of these sides 
is the tangent, which is determined for each point M, by calculat- 
ing the value of the subtangent PT’, or of that part of the line of 
abscisses, which is comprehended between the ordinate PM and 
the point 7’, the intersection of this tangent. The subtangent is 
determined in the following manner. 

Through the extremities MW and m of the infinitely small side 
JM m, we suppose the two ordinates MP and mp, to be drawn, 
and through the point JV, the line Mr parallel to AP, the axis 
of the abscisses. ‘The infinitely small triangle Wr m is similar to 
the finite triangle T’P.M, and gives this proportion, 

rms NM RIM ae TT, 
Now if we call 4P,x; PM,y; it is evident that Pp, or its 
equalr M, will be da, and rm will be dy; we shall therefore 
: te es PT Yada 
have dae d Tey eae ii tae 

This is the general formula for determining the subtangent of 
any curve whatever, whether the y’s and a’s are perpendicular to 
each other or not, provided that the y’s are parallel among them- 
selves. We shall now give an example of the application of this 
formula to any curve of which we have the equation. 

Let us suppose that the nature of any curve 4. were express- 
ed by an equation containing «, y and constant quantities. If we 
differentiate this equation, there can never be more than two 
kinds of terms, those multiplied by da and those by dy. It will 
then be easy, by the common rules of algebra to deduce, from 


Tangents and Subtangents. 29 


this differential equation, a value for dg? which shall contain on- 
ly terms of a, y, and constants ; by substituting this value in the 


iid 
dy 
gent in 2, y, and constants ; finally, putting instead of y its value 


formula i , or y X , we shall have a value for the subtan- 


in terms of x, deduced from the equation of the curve, we shall 
have the value of the subtangent expressed in terms of x and con- 
stant quantities only. So that to determine the position of this 
line for any point whatever JV, we have only to substitute, in this 
last result, in place of x, the value of the absciss 4P, which an- 
swers to that point. 

Suppose, for example, that the given curve is an ellipse, of 
which the equation is (4p. 112) yy = = (acx—zx*). Differen- 
tiating this equation, we have 


2ydy=— (adx—2edz) or 2a2 ydy=ab*dxe—2b?xdc; 


from this we deduce the value of aE by dividing first by dy and 


2ary 


ars dx 
then by the multiplier of dx, and find ir ea 5 sub- 
pe . ydax 
substituting this value in ‘dg? we shall have 
yada 2a? y? 


2 
finally, substituting for y?, its value _ (a « — w*) given by the 


equation of the curve, and reducing, we have 
ydx ep 2(ax—x?) _ax—x? 
dy a—2x da—x 
a value which is precisely the same as that which was found by 
Algebra (4p. 119), but which is obtained here in a more expe- 
ditious manner. 

We may observe here how this result justifies what was 
said (8) concerning the quantities which are to be rejected in 
the calculation; for, by employing here the differential calcu- 
lus, the rules of which, in this example, suppose the omission 
of the infinitely small quantities of the second order, by the side 
of those of the first, we arrive at the same result as in the Appli- 


> 


sd 


30 3 Differential Calculus. 


cation of Algeora to Geometry, where this subtangent was deter- 
mined in the most exact and rigorous manner. We see that by 
thus rejecting the quantities which were pointed out to be neglect- 
ed, we only impress upon the calculus the character which it 
ought to have in order to express the conditions of the question. 

We pursue a similar course in determining the tangents, sub- 
normals, normals, &c. 

Let us suppose, for the sake of greater simplicity, that the ab- 
scisses and ordinates (the z’s and y’s) are perpendicular to each 
other. In order to determine the tangent, we compare anew the 
triangle Mm r with the triangle 7'PM, and we have 

rm: Mms: PM.:- TM; 
but, by a property of the rightangled triangle, Mrm, we have 


—2 —2 
Mm= None Was “dy 3 
therefore, * dy: Ydx2+dy2 :: ¥: TM; 
y S/dx2 + dy ne Jd x3 ~~ db y2 
dy Vd y2 


de® dy? fda? 
=yJ d y? =) +1. 


Thus, after differentiating the equation of the curve, we shall 


therefore, TMs 


deduce thence the value of aa the square of which we substitute 


in this expression for the tangent; after which, putting in place 
of y its value in terms of w and constants, drawn from the same 
equation, we shall have the tangent expressed in terms of x and 
constant quantities. An application of this may be made to the 
equation of the ellipse, and the same value will be found as was 
formerly obtained (4p. 124); 

If the subnormal is required, we suppose the line MQ perpen- 
dicular to the tangent TVV, and observe that the triangles Mr m, 
MPQ, which have their sides respectively perpendicular each to 
each, are similar; we have 

MM ricitaan is 2 ie iP As ‘i 
that is to say, 
I ARN DTI 2 ah od 


x 


Then, after having eB ame the equation of the curve, we de- 


d 
duce thence the value fs which we substitute in one > and; 


Normals, Subnormals, &c. Ig 


‘completing the operation as before, we have the value of the sub- 
normal in terms of # and constant quantities. 


; 2 
In the ellipse, for example, the equation y? = * (a x — x), 
being differentiated, gives 


b2 
2 oie 
2ydy= "(adv—2eda); Mente: = a2 ‘inte 98 
a dav Big i : 

consequently the subnormal, 

ydy 6? .a—2”4 b? 

dx a nagar y sg ey 
as was formerly found (4p. 118). 

If the normal WQ were required, it might be found by com- 
paring anew the triangle Mr m with the triangle PQ. 

Let us take, as asecond example of the formula for subtangents 
and of that for subnormals, the equation of the parabola, which is 
yy=pu(fAp. 172). By differentiating, we have 
ee = eo and akg Daf 


2Qydy=padx; then a Pees a 


the subtangent, therefore, 


and the subnormal 
YO psi pys Dp 


cea | Sy Q 
which agrees perfectly with what has been already found (4p. 
180). 
We shall take, as a third example, the equation y™+"=a™ a", 
which is a general expression for parabolas of every kind. 
The name of parabola is given to every curve, whose equation, 


such as y™*" = a” a", has only two terms, but in which, the ex- 
ponents of x and y in the two members, have the same sign. 
By differentiating this equation, we have 
(etn) yt 9S d y oe nant ok gars 
from which 7 = Us pl : 
and the subtangent 
ydx _ (m+n)yrt?_ 
dy ae na™ zr ? 
and substituting for y” +7” its value a” a”, we find 


Fig. 4. 


32 Differential Calculus. 


yd (m -+ n) a™ x” _m+n i 

Ae eg Naa ER VR) in f . 
Whence we may see that the subtangent, in these curves, is equal 
to as many times the absciss a, as there are units in the exponent 
of y divided by the exponent of x. This holds true, as we have 
already seen, in the common parabola, where the subtangent is 
2 x, and where the exponent of y divided by the exponent of a, 
is in fact 2 or 2. 

Let us now'take, as an example, an equation the nature of 
which is expressed by an equation in terms of the differentials 
of the coordinates. Let us suppose, for instance, the curve BM 
( fig. 4) to be such, that the abscisses 4P, Ap, &c. being taken 
in arithmetical progression, the corresponding ordinates PM, p m, 
&c. would be in geometrical progression. This curve is called 
the logurithmic, because, while the ordinates represent successive- 
ly all imaginable numbers, the abscisses are their logarithms, and it 


will have for its equation AA 


= dx, since we have found (25) 


that this equation expresses the relation of numbers to their log- 


arithms. We shall have, therefore, a = “sand consequent- 


yd x 
ly the subtangent la 


for each point of the same logarithmic, the subtangent PT 
is always the same, and is equal to as many times the first 
ordinate “2B, or a, as there are units in the modulus m. 

30. When the equation of the curve is such that, x increasing, 
y diminishes, as in figure 2, then the line rm must be expressed 
by —dy (20); andthe proportion r. Wi: rm:: PM: PT, 
which serves to find the subtangent, becomes 

Ee en Yd © 
—dy:dx:i:y Id ear ig A 

Thus there will be no difference in the calculation; the enly 
change is that the tangent, instead of falling on one side of the .4, 
the origin of the abscisses, with regard to the ordinate PWV, will 


fall on the opposite side. ‘This is the reason why we may always 


will become ~ "4 


= am; that is to say, 


take 4 an as the formula of the subtangents; if the ordinates 
dy ; 5 


i a , sea ty 
decrease, the value of . —— will have a negative sign, which in- 


Tangent to the Hyperbola. 33 


dicates that this value must be referred to the side opposite to the 
origin of the abscisses. 

If, for example, we take the equation of the circle, the origin’ 
of the abscisses being at the centre, that is to say, the equation 
(lp. 103) being y? = 5 a? — a”, it is evident that, while Cp or 
x ( fig. 3) increases, y or P/M diminishes; so that the subtangent Fig. 8 
PT falls on the side of PM opposite to C, the origin of the ab- 


scisses. This is shown also by the calculus; for, if we differentiate, 
gb) 8S bhi meee 

we have 2ydy=— 2adz, and consequently evihics —!" then 
a 


d pees Hf — (i q? — x7? 7 : “ 

“ly - = rte Rates a value, of which the sign — in- 
: x 

dicates, that it should be referred to the side opposite to that 

yd x 


which is supposed in taking for the formula of the subtan- 


gent. 

Let us now take, as an example, the equation vy = a?, which 
belongs to the hyperbola between its asymptotes (4p. 163); we 
have yda + xd y= 0, and consequently, 

i — —~; therefore, RelA a ea lp a 
Ae ee fy ay y 
which shows that, to draw a tangent to the hyperbola considered 
between its asymptotes, we must take upon the asymptote nearest 
JM, the point in question (fig. 7), and on the side of PJM, which Fig. 7 
is opposite to 4, the origin of the abscisses, the line 
yal Mime 3) metho 2 

We see with what facility all these results are obtained by the 
differential calculus. ) 

In the same manner with parabolas of all kinds, we call by the 
name of Hyperbolas referred to their asymptotes, all those curves 
whose equation, such as y” = a" t" a—", contains only two terms, 
but in which the exponents of y and a, in the two members, have 
contrary signs. We leave these curves as an example for the ex- 
ercise of the reader. ‘The subtangent will be found to be 


m : “i i 
=——85 that is to say, that it falls on the side opposite to the 
2 


origin of the abscisses, and that it is equal to as many times the 
absciss as there units in the exponent of y, divided by the expo- 


nent of «x. 
5 


34 Differential Calculus. 


In general, we determine at once, by this calculus, all the sub- 
tangents, tangents, &c. of all the curves of the same family. 
Those are said to be of the same family, whose equation is form- 
ed in the same manner, and differs only in the magnitude of the 
exponent. ‘Thus we call by the common name of circles, all the 
curves in which any power of the ordinate is equal to the product 
of any two powers of the two distances of that ordinate from the 
extremity of the line a, on which the abscisses are taken. Their 
equation is y™t" =a” (a—a)", which comprehends the circle 
properly so called, when m=n=1. The equation 


aba pea 7 a (a aided x)" 
. . c 
represents ellipses of all kinds; and y"™t" = re (a+ x)", as 


well as y™t” =s (« —a)" belong to the family of the hyperbo- 


les. The figure of these curves is determined by means of their 
equation, as we have seen in conic sections (4p. 101); observing 
that y has only one real value when its exponent is odd, and two 
when it is even (2/9. 157); which determines that to the same 
absciss, there corresponds only one branch of the curve in the 
first case, and in the second, two, which fall on different sides of 
the same axis. 

When we know how to determine the tangents, normals, 
&c. we may easily solve the two following problems. Ist. From a 


given point without a curve, to draw a tangent to this curve. 2d. 


From a point given any where in the plane of a curve line, to 
draw a perpendicular to that curve. 
For the first of these, let us suppose that DWM (fig. 8) is the 


tangent required, st passes through the point BD. The gener- 


al value of PT or “<= will be easily found by means of the 


a 
equation of the curve. If 6D be drawn parallel to the ordinate 
PM, the line BD and its distance B42 from A, the origin of the 
abscisses, are considered as known, since the point D is given in 
position. Calling, then, DB,h; AB, g; AP, x; and PM, 7; 


we shall have 


BP =g¢ +2, nd TB = PT — BP =12* —2—g. 


Perpendiculars to a Curve. 38 


Now the similar triangles T'3D, TPM, give 
i Preys ees Le. 


That is to say, 


yda Re Uae ° 2, 0% a | 
ay —x“x“—gih:: rae Ur GY \ 22 rey 
udax hdx 


: ’ } dx 
Putting, therefore, in this equation, the value of ia drawn 


from the equation of the curve which has been differentiated, and 
then substituting for y its value in terms of « and constants, also 
given by the equation of the curve, and we shall have the ab- 
sciss x of the point of contact. And if the tangents, drawn 
through the point D, should be multiples, the last equation, in 
terms of », will give all the abscisses of the points in which these 
tangents must end. | | 

As to the case of the perpendicular, let us suppose DB ( fig. 5) 
to be the perpendicular required; the subnormal PQ will be 


vad (29) ; now the similar triangles DBQ and MPQ give 
DB BE PM PY, 


that is to say, employing the same denominations as above, 


Fig. 5 


IS ARNO MOR Dei dy 
he Sih e bea ata To orl Tas" 
therefore, 
hdy yd y 


am io Tae? 


an equation of which the same use may be made as in the pre- 
ceding case. 


The two solutions just given may be simplified by making the ax- 
is of the abscisses pass through the given point D, in a direction par- 
allel to its former position ; that is to say, by taking DK (jig. 8) for Fig.g 
the axis of the abscisses instead of TP, which only requires, if we 
make AW = z, and consequentiy, z = y —h, or y = z +h, that we 
substitute, both in the equation of the curve and in that of the prob- 
lem, z+ h instead of y We may also take the point D for the 
origin of the abscisses. 

When the curve has a centre, as the circle, the ellipse, &c. we may 
always suppose that the point 2 is in a diameter, and then the solu- 
tion becomes much more simple. 


36 Differentral Calculus. 


32. We may remark here, that i 


expresses the tangent of 

the angle which the curve makes, at each point, with the ordi- 
7 ae 

nate; and a is the value of the angle made by the element of 
ny 


f 
e 


the curve with the axis of the abscisses. For, in the right angled 


. triangled AZ rm (fig. 1), we have, supposing the radius of the 


tables = I, rm rds dos dang smh, 
Therefore tang rm M = pe a 
rin dy 


If then we wish to know in what place, a curve, or its tangent, 
makes with the ordinate a given angle, or one whose tangent is 
known, representing this tangent by m, we shall have 

oe ee 

cei 


so that, determining, by the differentiation of the equation of the 


m 


dx 
uy 
an equation, which, after having substituted therein for y its value 
in terms of x and constant quantities, deduced from the equation 
of the curve, will give the values of w answering to the points 
where the curve makes such an angle with the ordinate ; and if 
the curve no where makes with the ordinate an angle equal to the 
given angle, the value or values of a will be imaginary, or the 
equation will indicate a manifest absurdity. For example, in the 
hyperbola, which should have for its equation y? =2 (a @ +2), 
we should have 


curve, the value of , and making it equal to m, we shall have 


xan Qy 
dy” 2at4a’ 
which, being equal to m, gives 
Be i 
24+4e0° a+2u 
whence we deduce 


= Mt 5 


yr=mat2nnr; 
but the equation of the curve gives 
Y = V2 (ax x2) 5 
therefore ma+2mxr = /F(ax+22) » 
or, squaring both members, ; 
Mm? a2 14m? av + 4m?*e = 2a? x*, 
If itis asked, now, in what place this hyperbola makes with 


Rectilineal Asymptotes. 37 


the ordinate an angle of 45°; as the tangent of 45° is equal to 
radius, we shall have m = 1, which reduces the equation to 
a? +4a0+4a%7 =2ax 4-2 x? 


or 207 4+2an+a? = 0, 
which being resolved, gives 
es=—tatd/iat—ta? =—lat/—1@? ; 


thece values are imaginary, and show that the hyperbola, whose 
particular equation is 


i aaa? (a 2+ x2), 
no where makes with the ordinate an angle of 45°. 


Although the methods employed in this last remark and in the 
resolution of the two preceding problems, seem equally applicabie 
to the curves of which we have only the differential equation ; yet 
on close examination it will be found, that by this last equation the 
question cannot be perfectly satisfied by the calculus, as in the case 
where we have the equation in finite terms. Indeed, when the equa- 
tion, which resolves the question, still contains x and y, after the 


substitution of the value of a the differential equation of the 


curve will not serve to eliminate y, since, by the supposition, it con- 
tainsdwanddy. And evenif it should contain only a, we could 
not be sure that this value of # would satisfy the question. To be 
sure of this, it would be necessary to substitute this value of wx in 
the equation of the curve, and deduce thence a real value for y. But 
we cannot, in any way, deduce frem this equation, the value of y. 

The only means of resolving these equations, in such a case, is, 
supposing the curve already constructed, to construct also the equa- 
tion we have obtained, expressing the conditions of the question, 
which gives a second line, whose intersection or intersections with 
the first give the solution or solutions required. 

33. We may also, by the same principles, determine the rectiline- 
al asymptotes of curved lines. 

A curve has a rectilineal asymptote, when, having some branch 
which is infinitely extended, the tangent at the extremity of this 
branch meets the axis of the abscisses or that of the ordinates ata 


finite distance from their origin. ‘Thus (jig. 6) if, from the subtan- 
da j : da 
pent Pl S= has we take the absciss 4F =a, we shall have — x 


dy 
for the value of £7, or the distance of the origin 4 from the inter- 
section of the tangent. Alter having calculated, therefore, the value 
yd x 

of © 
dy 
by means of this equation combined with that of the curve, we may 
eliminate y or x, and we shall have s in termsof wor y. ‘Then sup- 
posing y or & infinite, if there result one or more finite values for s, 
they will be distances 4 C from the origin to the intersections of the 
axis and the asymptotes. But asa single distance does not fix the 


— x, we have only to represent it by a finite quantity s ; 


38 Differential Calculus. 


positions, we imagine, through the origin 7, a straight line 4K to be 
drawn parallel to the ordinates, and observe that the similar trian- 
gles TPM, TAK, give 
TP 2 PM ¢2TA': AR: 
that is to say, 
PRA ACL ks awdy 


Ge ALCRMEANE ICE Oot). eA Ota 
Kay wae dy Ci nAK = y a 


We therefore calculate, in the same way, the value of y — 


xdy 
dx 
and having supposed it equal to a finite quantity t, by means of this 
equation and that of the curve, we climinate x or y; and, supposing 
the remaining & or y infinite, the value or values of ¢ which result, 
will give the distances .2K3 which, with the distances £C, determine 
the position of the asymptotes. 

For example, if the equation of the curve were y3 = «3 (a+ 2), 
we should have 7 


Sytdy=lareda(a+ae)t+artda=2Qaxrdxu+3nx2 dx, 


> 


yd x sy 
then i hel —_—v = ee BT eon ATig 
dy Qax+3x 
or substituting for y° its value, 
yadx .. (Sen? +3 x8 ax? aac 


—_— 2 = 2 OF OE eS SSS 
dy 2anu +3 0% Qaetswt Bassa yy - 


then supposing » infinite, that is to say, neglecting 2a by the side of 


3 a, we shall have s = ie, We find, in like manner, 
J 
wdy Vax? +8 03. . 68.y3 — Zan? — Sx 
\ Hess aioes mrad 3 y? iti 3 y? ? 
which, if we substitute for y its value, is reduced to 
a x? 
baa AU AN A Oo gs 


3 
3. JS(a+ x22 
supposing then a to be infinite, we have t = 4a. 

Lf, the value of s being finite, we find that of ¢ infinite, it would 
prove that the asymptete is parallel to the ordinates. If, on the con- 
trary, s being infinite, t were finite, or zero, or infinitely small, the 
asymptote would be perallel to the abscisses. 

34. In all which precedes, we have supposed that the ordinates 
were parallel, and moreover that they all issued from the line on 
which the abscisses are reckoned. But it often happens that we 
make the ordinates issue from a fixed point. Sometimes we take as 
abscisses the arcs of a curve line, and as ordinates, straight or curve 
lines. But, in general, to whatever lines the points of the principal 
curve be referred, we always have or may have an equation which 
expresses the relation of tue abscisses to the ordinates. When we 
wish to make use of it to determine the tangents or other lines, we 
must take care that the lines we employ to determine these tangents 
contain no other differentials than those of the variables which enter 
into the equation of the curve. We shail now illustrate this by some 
examples. 


Drawing Tangents. 3 


85. Let us suppose, first, that AM (fig. 9), being a known curve Fig. 9. 
of which we know how to draw the tangents, 2S were a curve 
having fur its abscisses the arcs 4 of the first, and for its or- 
dinates the lines JZS parallel to a given line; the relation of 4M to 
JES being expressed by any equation, it is required to draw a tan- 
gent to the point S of the curve BS,° 

We imagine, as before, the infinitely small arc S's, of which the 
prolongation SQ, or the tangent, meets at Q the tangent M7 to the 
corresponding point AZ of the curve A.V; and, having drawn the 
line Sie parallel to #£T' or Adm, the triangle Sk s will be similar to 
the triangle Q.iZS, so that we shall have 

sk: Sk:: MS: MQ; 
now if we call the arc A.M, w ; and the ordinate AZS, y 3 we shall have 
Mm = Ske =dxe,andsk =sn—SH=dy; 


then ay ALS pM teal 


taking then upon the tangent AZT the part AZ equal to the value 


Be d ; sae 
of or ~ determined by the differentiation of the equation of the 


curve, we shall have the point Q, through*which and the point S, 
drawing @5, this line will be the tangent required. 

Let us suppose, for example, that the curve BS is described so that 
the ordinate ZS is always equal to a determinate part of the arc 
AJM; that is to say, that ZS is always to 4M in the given ra- 
tio of a to b, we should then have y : w :: @ : 6, and the equation of 
the curve is by =a. Dilerentiating, bdy == ada, and consequent- 


dx b yd x b } 
ly —— =-—-; then kA or MQ = tite 3 but by the equation of 
dy a dy a : 
b2 ; 
the curve, <1 = 2; therefore AYQ =a. Thus inall the curves 


whose parallel ordinates have always the same ratio to their corres- 
ponding abscisses, whether straight lines or curves, the subtangent 
QAM wiil always be equal to the corresponding absciss 442 

36. When the curve 447, upon which we take the abscisses, is a cir- 
cle; and the ordinate MIS is always to the arc AM ina constant 
ratio, the curve BS is called a cycloid. If the .ordinate JAZS is al- 
ways equal to AMM, itis the common cycloid, or that traced by a 
point in the circumference of a circle revolving ona plane. If MS 
is greater than A.M, but still having a constant ratio with it, it is 
called a prolate cycloid; if, on the contrary, ALS is less than AM, it 
is called a curtate cycloid. 

37. Lf the equation of the curve to which it is required to draw a 
tangent, instead of expressing the ratio of 0M to ZS, expressed 
that of Ai to PS; that is, if the arcs A.W were the abscisses xv, and 
the ordinates PES, or y, were reckoned froma determinate straight 
line 47; then Su being drawn parallel to 4P, the subtargent PI 
would be determined on the line 42, in the following manner. 

As the curve 4.47 is supposed to be known, the subtangent and 
tangent to each point of it, are also Considered as known; so that, 
making PT = s, and TM == t, we shall have, by drawing Mr par- 


Fig. 10. 


AD Differential Calculus. 


allel to 4P, and comparing the similar triangles 7PM and Mr m, 
TPT: Mrov Mais thaviays St 23 Mr dies 
therefore Mr = a 

Again ; the similar triangles Sus and 1PS give 
16S SSE ee eee eae ds 
andas PS is y,us is dy; therefore 
ds eer ; sydx 
dy : ; at Mi i Ae ay 
Then, differentiating the equation of the curve, we find the value of 


= which being substituted in ies will give the value of PJ, 
freed from differentials. 

38, Sometimes the equation of the curve is not given by the rela- 
tion of the abscisses to the ordinates, but by that which each ordi- 
nate of the curve is supposed to have to the corresponding ordi- 
nates of some other known curve. In that case the tangents are 
drawn by the following method. Let us suppose, for example, that 
the curve BMV ( fig. 10).depends on the two known curves 4H and 
ON, by means of an equation between the corresponding ordinates 
PL, PM, PN, which we shall call respectively x, y, z. Since the 
curves 4, and C.Vare supposed known, their subtangents PS and 
PR are also considered as known. Call PS,s, and PR, s’; and im- 
gine the ordinate px ml to be drawn infinitely near to PL, and Lu, 
Mr,no parallel to CAP. The similar triangles LPS, Lul give 

PS): PLiesiaest wmlssthatis; s ssauetieh bh dae 


= Su, supposing Su to be parallel to 4P. 


therefore Lu = ee = Mr. Now the similar triangles TPM, 
Mrmgiverm: Mr:: PM: PT; that is, 
SEE ed aaedapages apt ae SUE , 
dy: RB iN PT; therefore £ Bg 


then, if the equation of the curve contained only x and y, we should 


by differentiating this equation, have the value of a which being 


cer at thee d es 
substituted in : aye would give the value of PT freed from differ- 
xa 
entiats. But as this equation contains «, y, and z, its differential 
will contain d x, dy,and dz; we must therefore find the value of 
dz expressed in terms of dw anddy. New the similar triangles 
Vonand VPR give No: onor Lu:: VP: PR; that is, ob- 
serving that while PJM increases, PV’ diminishes, so that its differ- 
ence JV’ o or dz is negative, 
sdax : 
:3 23.8 3 thereforedz = -_, 
ww S Xx 
so that, putting in the differential equation of the curve, the quantity 
—szdx 


—_— 


> 
Sx 


—szda 
; Ree 


—dz: 


instead of dz, we shall easily find the value of = to be 


Drawing Tangenis. 41 


syd 
substituted in the formula Lo of the subtangent. As an exam- 


ple, let us suppose that 44 and CN being any two known curves, 
the ordinate P.M is always a mean proportional between PL and 
PN, wehavex: yi: yi x; orxz=y? for the equation of the 
curve BM. Differentiating, it becomes 

xadz+zdx=2Qydy; 


Aa ‘ szdx 
substituting for dz its general value — Feo we have 


i ia +zdx=2ydy, 
whence we deduce ae smc A 3; then PT = Ae becomes 
dy  z(s’ —s) addy 


'2ss/ y? ois 3 : 

—__—__—_—., or, substituting for y? its value x z,and reducing, 
xz (s' —S) 
Qss' 
S—Si 

It is easy to vary these examples by taking any equation we please 
in terms of x, y, and z We may, if we please, suppose 42 and 
CN to be straight lines (fig. 11). In this case, taking always PM Fig. 11. 
a mean proportional between PL and PN, the curve BA is a conic 
section, viz.a parabola, when the point C is infinitely distant, or the 
straight line C.V’is parallel to 4C; an ellipse, when the two angles 
HAC and HCA are acute ; and, particularly, a circle, when they are 
each 45°; and anhyperbola, when one of the two angles is obtuse. 

39 When the ordinates issue from a fixed point, we take as ab- 
scisses the arcs of a known curve, which most commonly is a circle ; 
that is, in this last case, the equation expresses the relation of the or- 
dinate CM (fig. 12) to the angle 40M, which that line makes with Fig. 12, 
anothe:. such as € given in position ; or it announces the relation of 
the ordinate CM to the are OS described with a determinate radius. 

To draw the tangents, when we have the equation between the or- 
dinate O.M and the angle CM, or the arc OS, we imagine that, for 
each point M, there 1s raised upon Ca perpendicular CT which 
meets the tangent 7M in T, then taking the infinitely small arc Mm, 
and drawing the ordinate m C, we conceive that with the radius CM, 
there has been described the arc Mr, which may be considered as a 
straight line perpendicular to Cm atr. As the angle Mm r differs 
infinitely little from the angle TWC, the triangles Mrm and TOM 
are similar, and give rm (dy) : Mr :: CM (y) : CT; or 

ote pe SCR 
Gap Ar, 33s OC pes da 

calling the are OS, x, and its radius, a, the similar sectors OS's and 
CMr give CS: CM :: Ss: Mr; that is, 
ydx 

al 
Substituting this value of Mr in that of CT, we have CT or the 


6 


I sod bog 


a:y::dx: Mr= 


Fig. 14. 


42 Differential Calculus. 


y2 dx 
subtangent = J 
a 


- Now as the relation between x and y is sup- 
posed to be known, it will be easy, by differentiating the equation 
which expresses this relation, to obtain the value of a which, be- 


ing substituted in CT, will give a new expression for CT, freed from 
differentials. 
If we suppose, for example, that the ordinate OM (jig. 13) is 


"always to the corresponding arc OS, in the ratio of m to n, that is, if 


yi“ i:m:n,wehave ny =m; then ndy = md 4, and con- 


ie eR 
quently aa therefore 


FY 3 
CT = Fe eer ey Bee 
ady a m a av 
; ny y x 
now, by the equation, we have = therefore CT = PES If, 


then, from the point C, as a centre, with the radius CM, we describe 
the arc MQ, we shall have CL = MQ. For the similar sectors COS 
and CQJi give CS: GS :: CM: MQ, that is, 


Csi 2 2 Us all ees a; therefore CT’ = AQ. 


The curve of which we have just been speaking is the spiral of 
Archimedes. 

40. Let us now suppose that OS (fig. 14) being a known curve, or 
one whose tangents may be drawn, the curve 8.M be constructed 
with this condition, that CS, x, and CM, y, have with each other a 
determinate relation expressed by a given equation. If we conceive 
the infinitely small arc Mm, the ordinates C.M, C m, and the arcs 
Mr, Sq, described from the centre € and with the radii CM and 
US, it is evident that the differentiation of the equation will give on- 
ly the relation of dy toda or rm tosgq, since y and & are the only 
variables which enter into this equation. But, in order to determine 
the subtangent CT, we must have the ratio of rm to rJfZ. Now 
y Jf may be thus determined by means of the conditions of the 
question 

Since the curve OS is known, the subtangent CQ for each point S 
is given ; now the similar triangles QUS, S qs give 

. OS): CQlscigs 210.5; 


calling, therefore, C Q,s, we shallhavex:s::da:qS= dics ; 
but the similar sectors CS g, CJM 7, give 
CS: CM :: Sq: Mr; ora: y 3: sida : My eae 3 
m2 A ee 


it is now therefore easy to find the subtangent CT, by comparing the 
similar triangles Mr m and TCM, which give 
rmirdf:: OM: OT, or dy: 29" ::4: oT = ‘ 
: x x2dy 
To apply this, let us suppose the curve BM to be constructed by 
taking SU always of the same- magnitude, or equal to a given line 


Limits of Curves. 43 


a, whatever the line OS may be. We ae have then, x7 +a= y. 
oY tg 


Therefore d a = dy, and consequently sai =: 1; the subtangent CT’ 
2 
becomes then CT’ = a which is constructed in the following man- 


ner. | 

Through the point Mand parallel to the tangent SQ, we draw a 
line MX; then having joined the two points S and X, we draw from 
the same point Ma line MT parallel to SX, and MT, thus deter- 
mined, will be the tangent required. For the similar triangle CSQ, 
OMX, give CS: CQ:: CM: CX, or wisiiy: CX= 
In like manner, the similar triangles CSX and CMT, give 


2 2 
CS: OX:: COM: CT; ox: —Ls:y: OT ==4. 
4 be a 


We perceive, by these examples, how to proceed in the application 
of these methods to other cases. It is to be observed, in conclusion, 
that when OS's a straight line, the curve BJ, which is formed by 
taking SW always of the same magnitude, is what is called the 
Conchoid of Nicomedes. 


Application to the limits of curve lines, and in general to the lim- 
ats of quantittes, and to questions on Maxima and Minima. 


41. We have seen (32), that os expresses the tangent of the 


angle which the curve, or its tangent makes, at each point, with 
é di : 
the ordinate ; and that io represents that of the angle which 
x 

the curve or its tangent makes with the axis of the abscisses. 

To know, therefore, at what point the tangent of a curve be- 
comes parallel to the ordinates, we must find the the values of « 

dx 

and y corresponding to a 0, or simply to dx =03 and to 
find where the tangent of the curve is parallel to the abscisses, 


we must suppose oe = 0, or merely d y = 0, and the values of « 


and y which result, are those of the point of contact. 

If evidently follows from this that in order to find whether a 
curve, of which we know the equation, has, at any point, its tan- 
gent parallel to the ordinates or to the abscisses, it is necessary 


to differentiate the equation, and having deduced the value of a 
y 


if we make the numerator of this ratio equal to zero, we shall 


Vig. 15, 


44 | Differential Calculus. 


have an equation which, with the equation of the curve, will give 
the value of x and that of y, which determine at what points the 
tangent is parallel to the ordinates; so that if these points are 
multiple, we shall have several values for x and y. 

On the contrary, if we make the denominator equal to zero, 
this equation, conjointly with that ofthe curve, will determine 
the values of # and y, which answer to the points where the tan- 
gent of the curve becomes parallel to the abscisses. It must 
however be observed, that although dw is always zero, when the 
tangent is parallel to the ordinates, as well as dy, when the tan- 
gent is parallel to the abscisses, still it must not be concluded, 
when the value or values of 2 resulting from the supposition that 
d x = 0, or that d y= 0, are found, that the tangent is parallel to 
the ordinates or abscisses, except when we have not at the same 
time dw =0, anddy=0. 

To illustrate these rules by a familiar example, let us take the 
curve which has for its equation 

y? +e? =3ax—2a2+4+2by—)?, 
which, on the supposition that the a and y are perpendicular to 
each other, belongs to the circle, the origin of the coordinates 
being at the point 4. 

The lines AP (fig. 15) are x, and the lines P/V, PJW’ are the 
two values of y which the resolution of the equation gives for 
each value of a. 

If we differentiate this equation, we have 

2Qydyt+2Qeadxr=sadx+2bdy, 


whence we deduce 


Let us first make the numerator equal to zero; to find the 
points at which the tangent becomes parallel to the ordinates, 
we shall have 2y— 26=0, ory=6. Substituting this value 
in the equation of the curve, it becomes, 

b2 12% = 3ax—2a? +262 —b?, orxz? —S3axr=—2a?, 
which, being resolved, gives 7 =3a + 4/1a? ; thatis,aw=2a, 
and #=a; which shows that the curve or its tangent becomes 
parallel to the ordinates at the two points # and R’, which have 
for the ordinate the line 0, and of which one, A, has for its ab- 
sciss the line 4Q =a, and the other, R’, has for its absciss the 
line 4Q’ = 2a. 


Limits of Curves. 45 


I ait 
Let us now make the denominator of aa equal to zero, to find 


at what point the curve or its tangent becomes parallel to the ab- 
scisses. We shall have 3a —2a=0,ora=ga. Substituting 
this value in the equation of the curve, we have 
-y? 42a? = $a?—2a24+2by— 6?, 

or y> —Q2by+b%=14a7; 
and, extracting the square root, y— 6 = + 14a; therefore 

y=b+14a; thatis, y=b+4a, and y=b—ia4, 
which shows that the tangent becomes parallel to the abscisses, 
at the two points T' and I”, which have as their common absciss 
the line 4S= $ a, and of which, 7” has for its ordinate 

ST =b+24a4, and T, the line ST'=b —ia. 

The points @ and @’ are called the limits of the abscisses, be- 
cause, between @ and Q’, for each absciss 4P are correspond- 
ing real values P/M and PM’ for y; while between Q and 4, 
and beyond @ with reference to 4, there is no point of the curve, 
so thatif x be supposed smaller than 2Q or a, or greater than 
1’ or 2a, no real value can be found for y. Indeed, if in the 
equation we substitute for 2 a quantity a — gq smaller than a, or 
a quantity 2 @-+q greater than 2 a, on resolving the equation, the 
two values of y will be found to be imaginary. 

In like manner, if through the point 4 we conceive AL’ to be 
drawn parallel to the ordinates, that is, to the axis of the ordi- 
nates; and if through the points 7’ and 7”, the lines 7'L, T"L' be 
drawn parallel to the abscisses; the lines 

4L= ST=b—ia, and AL’ = ST’ =o +14, « 

are the limits of the ordinates; for.it is evident that there can be 
no ordinate greater than JL’, nor smaller than 42, the tangent 
being supposed parallel to the abscisses. In fact, if in the equa- 
tion of the curve, a quantity, 6— 4a—q smaller than — 1a, 
be substituted for y, it will be found, on resolving the equation, 
that the values of ware imaginary. ‘The same thing will happen, 
if instead of y, the quantity 6+4a-+4q, greater than 6+4a be 
substituted. 

42. The ordinate ST” is the greatest of all those which termi- 
nate in the concave part RT'#' of the circumference. The or- 
dinate S7" is the least of those which terminate in the convex 
part; and the ordinates QR, QI are, at once, the least for the 
concave part, and the greatest for the convex. 


A 


46 Differential Calculus. 


43. Thus the same method serves at the same time, Ist, to as- 
sign the limits of the abscisses and ordinates; 2d, to determine 
in what cases the tangent becomes parallel to the abscisses or to 
the ordinates ; 3d, to find the greatest and the least abscisses or 
ordinates. 

44. Now in whatever manner a quantity be expressed algebra- 
ically, the algebraical expression which represents it, may always 
be considered as the expression for the ordinate of a curved line. 


M7 X(A—X) . : ; 
For example, if st ed is the expression of a quantity 


x? x (a— zx) 
az 
consider this equation as that of a curve line whose absciss is @ and 
x? x (a— x) 
2 


which we call y, in which case we have y = , We may 


ordinate y. If then the quantity may, in a certain 


case, become greater or smaller than in any other case (which is 
called being susceptible of a maximum or minimum), it is evi- 
dent that we must pursue exactly the same method as above, that 
is, differentiate the equation, and having deduced from it the 
dx 
dy 
equal to zero. 

45. It is to this that the method which is called that of maxima. 
and minima is reduced. ‘This method is one of the most useful 
in analysis, and has for its object to find, among several quantities 
which increase or decrease according to a certain law, that which 
is greatest or least; or, in general that, among all similar quanti- 
ties, which possesses certain properties in the highest degree. 
We shall now give some examples from Geometry and Algebra. 
Mechanics will hereafter furnish some at once more curious and 
more useful. 

46. Let it be proposed to divide a given number} a into two 
parts, such that their product shall be a maximum. Call @ one of 
the parts ; the other will be a — 2, and the product will be repre- 
sented by aw—a*. Let this product = y ; we have y=ax—2a?; 
by differentiating, therefore, dy=ada—2adza, and conse- 
od 
a—2x 


value of —--, make the numerator or denominator of that value 


dx ; 
quently —— = ; if we make the numerator equal to zero 
dy ; 


+ See the Introduction. 


Maxima and Minima. 47 


we have |! =0, which is absurd; consequently if there be a 
maximum, it will be shown by making the denominator equal to 
zero; let therefore a — 22 =0, whence x =a, which shows, 
that among all the different ways in which a number may be di- 
vided into two parts, the product of the two parts will be greatest, 
when each is one half of the number.t 

47. When, as in this example, we have the algebraical expres- 
sion of a quantity. we may dispense with making it equal to a new 
variable y; we have merely to differentiate, and make the nume- 
rator or denominator, if the differential is a fraction, equal to ze- 
ro. ‘Ihus, in the same example, we merely differentiate aw — x”, 
and, making the differential adw—2xd«a equal to zero, we. 
have ad x — 22d x=0, whence we deduce, as before r= a. 

48. T’o take a more general question, let it be proposed to di- 
vide a known number a into two parts, such that the product of a 
determinate power of one of the parts by the same or another 
power of the other part shall be the greatest possible. Let x re- 
present the first part, and m the power to which it is to be raised ; 
the second part will bea —ar, and if nm designate its power, the 
product in question will be z™(a— wx)". If we differentiate this 
product and make the differential equal to zero, we shall have 

mx™—1 da (a—x)"—na”dx(a—ax)"-! =0. 
Dividing the whole by a~' d x (a—2)"—', we have 
m (a—x)—nx=0, orma—mex—nx=O0, 

ma 
n+n 
~ required to divide a into two such parts that the square of one 

multiplied by the cube of the other part, shall be the greatest 

product »ossible; then m=2,n=3. We have therefore 

2a 
243 
that is, that one of the parts must be 2 of the number or quanti- 
ty proposed, and the other be consequently three fifths. 

What has been said above in relation to figure 15, shows that 
a quantity may become greatest among all quantities of the same 


which gives « = . If we suppose, for example, that it is 


pit’ $9) ° 
= 345 


— 
— 


t+ It is here to be observed that those considerations employed in 
the solution of this question in the introduction, do not here appear, 
but are contained in the calculus, being principles which are at the 
foundation of this mode of considering questions. 


Fig, 16. 


48 Differential Calculus. 


kind, in two different ways ; either when increasing first, like PM’, 
it diminishes afterwards, or when, like P’ VW” it continues to in- 
crease, until it stops suddenly on becoming Q' R’; but in this 
last case, it is at once the greatest of all the ordinates which ter- 
minate in the convex part, and the least of those which terminate 
in the concave part. In like manner a quantity may become least 
of all those of the same kind, in two different ways; either, like 
P.M, by diminishing at first to increase afterwards, or, like P’M’”’, 
by diminishing until it suddenly stops, and then it is at once a 
minimum and a maximum ; it is a minimum with reference to the 
branch MT’, and a maximum with reference to the branch 
MTM". 

49. So that, to ascertain whether a quantity is a maximum or a 
minimum, or both, we must, supposing a to mark the value of a, 
corresponding to the maxumum or minimum, substitute successive- 
ly instead of x, in the quantity proposed, a + q,a, anda—q. If 
the two extreme results are real and less than the middle one, the 
quantity is a maximum ; if, on the contrary, the intermediate re- 
sult is the least; the quantity is a minimum ; finally, if of the two 
extreme results, one is imaginary and the other real, the quantity 
is at once a maximum and a minimum. 

50. When, in the determination of a maximum or minimum, the 
value found for the variable renders that of the maxymum or mini- 
mum negative, we must conclude that the maximum or minimum 
which it indicates, does not belong to the question under consid- 
eration, but that it answers to a question in which some of the 
conditions are of a contrary character. If, for example, it were 
required to divide the line 46 (fig. 16) at the point C, in such a 
manner that the square of the distance 4C from the point 4, be- 
ing divided by the distance BC from the point B, gives the least 
possible quotient ; then, calling the given line 22, a, and the dis- 
tance 4C, x; the remainder CB isa—a, and consequently the 


quotient is — 
tu— 


= differentiating this quantity or x? (a—«)-}, 


we have 2vedx(a—2X)~'+2?dx(a—zx)-*=0, 


2 oe we x dx £56 
= a—«ec ' (a—wxe” ? 
or Qaxdx—x*dx=0, or (La—ax#)r=0; 


which gives either «= 0, or 2a— x= 0; the first value indicates 


Maxima and Minima. 49 


a minimum, which is evident without calculation. As to the sec- 


2 
- . . . . . Xx 
ond, which gives c= 2a, if we substitute it in peepee find 


ee 
f 


4a? 
ee 
present question. But if we examine the value of x= 2a just 
obtained, we see that the point C cannot be between 4 and B, 
but that the question will have a solution, if it be required to find 
itin 46 produced beyoud B with regard to 4. In that case, if 
we call 4C’, x, the distance BC’ will not be a—a, but 2 —a, 
ue 


or —4a. The minimum, therefore, does not belong to the 


and the quantity under consideration will be 


, which, being 


X— Ot 
differentiated and made equal to zero, gives 


Qudx x2 dx 
AE ites a 


or, after making the reductions, 

a= dx—2arcdzr=0, 
which gives a= 2a, as before ; but this quantity, substituted in 
2 


x 


“5 changes it into4a. There is therefore a minimum for this 


case. 

If the denominator x — a of the differential be made equal to 
zero, it is ea, which indicates a maximum ; and indeed, when 
«% =a the quantity becomes infinite. but it has nevertheless the 
true character of a maximum, for whether we suppose @ greater 
or less than a, it gives a less quantity than to suppose 7 =a. 

51. When the expression of a quantity of which we wish to 
know the maximum or minimum, contains any constant multiplier 
or divisor, we may suppress this multiplier or divisor before dif- 


aikts ° at : 
ferentiating ; for if we suppose that +" represents a quantity 


which is to be a maximum, or a minimum, a and 6 being constant, 
dy 


must be = 0; but since a and 6 are not zero,dy must 


necessarily be =0; the conclusion is therefore the same as if the 
maximum of y only_were required, that is, the same as if the con- 
stant divisors and factors were suppressed. ‘I‘his remark serves, 
in many cases, to simplify the calculation. 
52. Let itaow be proposed to find, among all the lines which 
may be drawn through the same point D given in the angle 
7 


50 Differential Calculus. 


- ABC (fig. 17), that which forms with the sides of the angle the 


least possible triangle. 


Through the point D let DG be drawn parallel to the side 4B, 
and supposing Hany straight line drawn through the point D, 
let fall upon BC the perpendicular DK, and from the point £, 
where EF meets 2B, let fall upon BC the perpendicular EL. 
The line BG is considered as known as well as the perpendicu- 
lar DK; let therefore BG = a, DK = 6, and let B&, the base of 
the triangle BEF, be =x. We see that, from a certain point, 
the more BF increases, the greater will be the triangle. If, on 
the contrary, B& diminishes, the triangle will diminish also, but 
only to a certain point; for if BEF should become nearly equal 
to BG, the straight line EDF would be nearly parallel to AB, 
since it would nearly coincide with GD, in which case the trian- 
gle would be very great. There is then a certain value of BF, 
which gives the smallest triangle possible. In order to find this 
value, we must discover the general expression of the triangle 
BEF. Now the similar triangles BEL’, GDF give 

GPs (BE eit DE cE Es 
and the similar triangles DKF and ELF, give 

IDB ye ERE) 22 BIC Fads 5 
therefore GF: BF :: DK : EL; that is, 


e@—a:nv::6: EL=— ee : 


—a 
ELXBF .. 
therefore the surface of the triangle BEF, which is sbeot | will 
be ee Fil x = or 2 eee We must therefore differentiate this 
x— a a 


quantity, and make the numerator or denominator equal to zero, or, 
since we may suppress the constant factor 45 (51), we need only 


x2 
differentiate the quantity — ge and, not to repeat an chee 


which has been already Me (50), we shall find a= 2a; if, 
therefore, we take BF =2a=2 BG, the line FDE, drawn 
through the point D, will give the triangle PLE for the minumum 
required. 

53. Let it be proposed to find, among all the parallelopipeds 
of the same surface and the same altitude, that which has the 
greatest capacity. 


Maxima and Minima. 51 


Call h the altitude and c? the surface of the parallelopiped, x 
and y the two sides of the rectangle which serves as base. The 
whole surface is composed of six rectangles, of which two have 
each x for their base and fA for their altitude, two others have h 
for their altitude and y for their base, and the two last have w for 
base and y for altitude ; so that the whole surface is expressed by 

Qha+2hy+2ry; 
that is, we have 
Qhe+2hy+2ry=c?. 
The capacity or solidity is hwy. Since then it must be the great- 
est of all those of the same surface, we have hady +hydx=0, 
or, what comes to the same thing, 
vcdy+ydx=0, 
But the equation 2ha+2hy+2x«y=c?, which shows that 
the surface of all these parallelopipeds is constant or the same, 
gives Qhdx+2hdy+2rdy4+2ydx=0. 
Substituting, therefore, in this equation, the value of d@ found in 
the other, we have, after making the reductions, y =a; the base 
therefore must be a square. To find the value of the side, we 
substitute for y its value x in the equation 
Qhe+2hy+2xry=c?, 
which thereby becomes 4h x + 2 x? = c? ; which, being resolved, 
gives c=—A+yi2, $02, whose root r=—h—y h? +402, 
being negative, does not belong to the present question ; thus the 
true value of x is / 
e=—h+Vi2{ her 

54, If itis now asked what must be the altitude A, in order 
that the parallelopiped may have the greatest solidity among all 
those of the same surface ; we observe that since, the altitude be- 
ing A, the base must be a square, this solidity will be expressed 
by hax? ; considering then A and @ as variable, the differential of 
hx? must be equal to0; we have therefore 

Qhadx+22?dh=0,or2hdr+ardh=—o. 
But the equation 4h x + 2 x? =c?, which indicates that the sur- 
face is constant, gives for its differential, 
4hdx+4ardh+4rdr=0; 
substituting, therefore, in this equation, for dh, its value found 
from the equation 2h dx +adx=0, and making the reductions, 
we have h= «a; the parallelopiped sought must therefore be a 


52 Differential Calculus. 


cube, since its altitude h is equal to the side x of the square 
which serves as base. ‘To find now the value of the side of this 
cube, we must substitute for 4 its value v, in the equation 
Aa ft, Canta Qin sae) 68 | 
which thus becomes 
Awan 2p? CON Outs => 02, 

which gives « = [2 

Therefore, of all the parallelopipeds of the same surface, that 
which has the greatest solidity is the cube which has for its side 
a line equal to the square root of the sixth part of that surface. 

55. Let it now be required to find, among all the triangles of 
the same perimeter and same base, that which has the greatest 
surface. 

Let a be the base 4B, and c the perimeter of the triangle 


ABC (fig. 18). Let fall the perpendicular CP, and call AP, x 5 


CP, y3; we shall have 

PB=a—2; AC= 172-32, 
and CB = vVy2 +(a —2)2 
Then the perimeter will be 


Vxr py? + /y2 + (a—2z)2? +Aa=C, 


and the surface => ji # 
Now, by the conditions of the question, we must have 
ae =o, ON, Y ots 

But, if we differentiate the expression for the perimeter, we have 

xdetudy —dx(a—) +ydy hy 

Vx8 + y3 VS 92 + (a—2)3 
which, dy being = 0, is reduced to 
xd x (c—av)dx 

* re {) 3 


or, dividing by da, and freeing from fractions, 
@ 4 /y2 + (a—ax)2 — (€A— &) 442222 =0. 
Squaring the two members, performing the operations indicated, 
suppressing the terms which cancel each other, and reducing, we 
arrive at this equation, 
xn? =(a—a2)? =a? —2ar+2’; 
whence « =a, which shows that the triangle must be isoceles. 


Maxima and Minima. 53 


To construct it, we raise a perpendicular from the middle of AB, 
and having described from the point B as a centre, with a radius 
equal to half the excess of the perimeter c over the base a, an 
arc cutting that perpendicular in C, and drawn CB and C.A, we 
have the triangle which has the greatest surface among all those 
of the same perimete”, and the same base. 

56. If it is now required to find, generally, among all the trian- 
gles of the same perimeter, that which has the greatest surface, it 
must be observed, that whatever be the base, we see by the pre- 
ceding solution that x must always be equal to half of it, that is, 
that whatever be the values of a, wis always =ia. This being 
the case, the equation which expresses the perimeter will be re- 
duced to / 1a? + y? + fia +y? -a=¢, 
or 2°/iv+_y=c—a; 
squaring and finding the value of y, we have 


_ Je®—2ac 
he Gris odie 


The surface —* of the triangle will therefore be 


a c2 —Zac 
2 4 : 


Since then it must be the maximum among all those of the same 
perimeter, whatever may be the base a, we must make equal to 
zero the differential of 


2 
a c* —2eec pe RO, 
ea or of a/c? —2a¢, 


« being considered as variable. We shall have then 


d.aya@adae =d.a(c®—2ac)? 


i ae | 
= (c? —2ac)?da—cada(c*? —2ac) 2 
cada 
= d aa/c2 —2a¢ —n ey eae pom 


Making the denominator to disappear, 


da(c? —2ac)—cada=0; whence a= = ; 
therefore the base a must be athird of the perimeter; and as we 
have already found the triangle to be isoceles, it follows that it 
must be equilateral. Therefore of all the triangles of the same 


perimeter, the equilateral is that which has the greatest surface. 


54 Differential Calculus. 


57. In these two solutions, we have not made the denomina- 
tor equal to zero, because, in the first, that would have given an 
imaginary value for 2; and in the second, we should have found 
a =, which would have no better satisfied the question, since, 
if the base were half of the perimeter, the two other sides would 
be confounded with the base, and the triangle would be zero. In 
future, whenever the supposition of the numerator or denomina- 
tor made equal to zero, would lead to no admissible solution, we 
shall, to avoid useless investigations, pass over it without notice. 


58. In the last question but one, we have been able to determine, 
among all the parallelopipeds of the same surface, that which has the 
greatest capacity, only by first considering parallelopipeds of the same 
altitude. In like manner, in the last question, we have found among 
all triangles of the same perimeter, that, which had the greatest 
surface, by first resolving the question for triangles of the same base. 

It is usually more simple to proceed in this way; that is, to re- 
solve the question by making the least possible number of quantities 
vary at once, and afterwards successively treating as variable each 
one of the quantities which have been considered constant.  [f, for 
exumple, it were required to divide a given number into three parts, 
in such a manner that the product of these three parts should be 
the greatest possible ; calling a and y two of these parts, and a the 
given number, the third part will be a—-.«—y, and the product of 
the three will be wy (a —«— y), of which the differential must be 
made equal to zero. But, instead of considering x and y as both va- 
riable at the same time, we shall differentiate, considering x only 
as variable; we have then, 

ayda—2xydx—y? dx =0, 
whence we deduce x= 1(a—y). 
The product x y (a —w# — y) is thus changed into Ly (a —y)?. We 
now differentiate, considering y as variable, and have 

sVYG—y)? oy dy (a —y), 
which we also make equai to zero, and have 

dy (a—y)? —2ydy (a—y) =0, 
whence we deduce yt as 
wherefore x, and a— a — y are each equal to 4a. 

59. We may also, if we please, make all the variable quantities 
vary together, then, collecting all the terms which are multiplied by 
the differential of the same variable, make their sum equal to zero, 
and do the same thing with regard to the differential of each variable. 
Thus, in the last example, we should have 

axdy+-ayda—20uyduex—axr-dy—2Quydy—yrdx=0, 
whence we obiain, by making equal to zero the sum of the terms af- 
fected by d x, and thai of the terms affected by d y, 

aydx—inyda—yrdx =), 
and axdy—zxexydy—xdy=0; 


Maxima and Minima. 55 


or, dividing the first by y dx, and the second by x dy, 
a—2y—x=0, 

equations from which we immediately conclude that « = 2a, and 

that y = 4a, as before. 

It is easy to perceive the reason of this process, if we observe that 
the only condition to be answered is that the whole differential be 
equal to zero. Now this condition can be answered generally in one 
of two ways only, either by supposing each of the two differentials 
dx and dy equal to zero, which indeed satisfies the equation, but 
makes nothing known, or by supposing that the sum of the terms 
multiplying dw, and of the terms multiplying dy, are each zero, 
which is precisely what we have done. | 

60. When the conditions of the question are expressed by several 
equations, we must, before applying this rule to the differential equa- 
tion which is to determine the maximum or minimum, deduce from 
the other differentiated equations the values of the differentials of as 
many variables as there are equations besides that, and substitute them 
in that equation ; the rule is then to be applied as if there were that 
equation alone. ‘hus, in the example already given of the greatest 
parallelopiped, we had this equation, 

Qhx+2Qhy+2X%uy=c?, 
and the condition that hay was to bea maximum. If then h, «x, 
and y are to be considered as varying at once, the first equation, be- 
ing differentiated, gives 

Qhdx+2uwdh+ 2hdy+2ydh+2undy+2ydu =0, 
and the condition of the maximuin gives 

hadythydx+aydh=0. 
From the first we find 
whe —ydx—xdy—hdy—hdx | 
= eae ; 
substituting this value in the second equation, we have, after the 
usual reductions, 


hx?dythyrdx~—xy?dx—a* ydy=0. 
We may now put equal to zero the sum of the terms multiplying d x, 
and that of the terms multiplying dy. We have 
hy?—xy? =0 orh =a, andh a? —ax2* y=O0orh=y; 
and since h = a, we have also y =; the three dimensions a, y, 


and h are therefore equal, which agrees with the former solution ; 
and putting these values in the equation 


Q2Qhao-+2hy + 2y x =c?, 
we have 6h? = c?, whence 


Aas cc 
= |-a- 
as in the former solution. 
61. We may not only make the quantities vary successively, or 
all at once, but we may take as constant any function whatever of 


Fig. 19. 


56 Differential Calculus. 


these quantities, provided that the number of these new arbitrary 
conditions, united to that of the condifions of the question, be not 
greater than the number of the variables z, y,z, which erter into the 
question. ‘This remark may be of the greatest use in many ques- 
tions, especially when there are radical quantities For example, 
let it be required to find among all the quadrilaterals of the same 
perimeter, that which has the greatest surface If from the angles C 
and D (fig. 19) we let fall the perpendiculars CE and DE upon the 
side 2B, and from the point D draw Dé& parallel to 4B; then call- 
ing AE,s; DH,t; AF,u; CF,x«; and BF,y; we have by the 
property of the rightangled triangles, 
then if the perimeter be equal to a, we have 

Jie + VePo? Per)i + Vp tuty=a, 
Again, the surface 4BCD is equivalent to the trapezium 
, DEFC — the triangle DAE + the triangle CFB; 
that is, 


aBed =(—*) 6+ — F424, 

This being laid down, it would be necessary to differentiate the 
two preceding equations. But the radical quantities would render 
the subsequent operations very complicated. To avoid these diffi- 
bulties, we suppose, at first, that the three radical quantities are con- 
stant, which gives 

haa 3 -1 gsds+idt 
d (/s2 +2 Vise 0 (624-45) Sd (8? 02) (6?) 
V/s2 +12 
which being = 0, because the quantity was considered as constant, 
we have 
Ist, sds +tdt=0. 
We find also for the differential of the second radical 

2d, (s+u)(ds+du) + (w—t) (dx—dt) =0; 

and for the third 
3d, xdx+ydy=0. 

The equation of the perimeter being differentiated on the same 
supposition, gives (since the differential of the radical quantities sep- 
arately is zero), 

The expression of the surface being differentiated and put equal to 
zero on account of the condition of its being a maximum, gives, 5th, 
(s--u) (dt+d x)-+(t-+a) (ds+du)—tds—sdi+idy+ydx=0, 
or 
udt+sdxa+uda+tidu+tawxdst+adutadytydx=o0. 

—tdt é | 
The first gives ds = =" ; the third da =— ee and the 


fourthdu—=—dy. Substituting these values in the second and 


fifth, we have, after all reductions, 


—(tdt+sdy)(u+ts)a—(ydy+adt)(a—t)s=0, 


Maxima and Minima. 57 


and 
suxdt—suydy—s? ydy—tsuxdy—ax* tdt—sy? dy=0: 
If we deduce from this last the value of dt, we shall see that all the 
terms of its numerator are affected by s, and that, by substituting it 
in the preceding, all the terms will also be affected by s, where- 
fore s = 0, which shows that the angle YB must be a right angle. 
This being determined, the equation of the perimeter is reduced to 
t+ Veqe i + Verte buty =a; 
and the expression of the surface becomes 
u ary 
Let us now differentiate, supposing only the two radicals constants 
we have | 
udu + (x—t) (dax—dt)=0; 
xde+tydy=0; 
u(dt+tdx)+ (¢+u)dutady+ydx=0. 


The second equation gives 


the third, | 
dt =—du—dy = (substituting for d y) we 
these values, substituted in the first and fourth equations, give 
yudu+ (w—t) (ydx—xdax+tydu) =0, 
and 
u(adx—ydu+yde) + (t+ a)ydu—axedr+yda=8. 
Now, if, from one of these we find the value of w, and substitute it 
in the other, we shall have an equation of which all the terms will 
be affected by y, and which consequently gives y = 0, and shows that 
the angle CBA must also be a right angle. ‘This being the case, the 
equation of the perimeter becomes 


t+ J/ut+(e@—)% +L pU=a, 
and that of the surface = (t + x) X > 


differentiating, therefore, supposing the radical constant, we find 
udu + («—t) (da —dt) =0, 
dt+du+dxa=0, 
u(dt+dv)+(¢+a)du=0. 
The second gives dt =—dx—du, 
and substituting this value in the two others, 
udu+t(a—t) (Qdx+du)=0, 
and —udu+(t+a)du=0. 
This last gives du = 0; whence the preceding becomes 
(w—t) 2d x =0, whence x =t. 
| 3 | 


58 Differential Calculus. 


This being determined, the equations of the perimeter and surface 
are reduced to 2t + 2u =a, and suriace = tu. 

The lines 4B, 40, BOC, CB, are all equal ; and since the angle 4 
must be a right angle, th® other angles being such, the quadrilateral 
sought is a square. 

We might have arrived at this property more readily, but that was 
not our principal object Ve wished to show how the liberty of 
treating certain quantities as constant, may, in many cases, much fa- 
cilitate the operation, and this example was well adapted to the pur- 
pose, as, without this artifice, the calculation would have been very 
complicated ; similar ideas may be applied to the other polygons, and 
it will be found, that, in general, of all the polygons of the same 
number of sides, that is greatest which is a regular polygon Whence 
it follows, that, of all figures of the same perimeter, the circle is that 
which contains the greatest space. 


Of Multiple Points. 


62. We have examined what takes place when one of the two dif- 
ferentials dw or dy, or which is the same thing, when the numera- 


4 eyo har 

tor or denominator of the fraction Ta 
ay 

found that one of these two cases always exists whenever there Is a 
maximum or minimum. But it may be asked what takes place, if the 


becomes zero; and we have 


t : dx 
denominator and numerator of the value aa become zero at the 
; : : dx A 
same time, and to what, in that case, is the value of rae reduced ? 
a 
In answer to these questions, we first observe, that when we differ- 
entiate the equation of a curve, as there are only terms multiplied 
by dw and those multiplied by dy, we may, calling 4 the sum of the 
former, and B the sum of the latter, represent the differential equa- 
: Sarath ; . dx 
tionby 2dx-+ Bdy=0. This equation gives i Can pol 
¥ 
in order that 4 and B should become zero at the same time, they 
must have a common divisor, which, becoming zero when w and y 
have certain values, renders B and # equal at the same time to zero. 


: ‘ , y x(a—2x)? 
For example, in the curve which has for its equation y? = Ghost 


at 
dx ‘ 9a y 
we have —— = -—— : 
dy (a — vw)? — 24 (a— x) 
or, substituting for y, its value, 
d e Qa (a — 2x) = 
Gare, —- ‘ ; - 
ay (@— x)? —22(a—a)’ 


a quantity which becomes $, when x =a; but we see, at the same 
time, that w— a 1s a Common divisor of the numerator and the de- 


; rhs 
nominator, and that the value of Tae reduced to 


Multiple Points. 59 


x 
ax eds wn Are 
dy qin 8 a7 
which, in the case of w =a, is reduced to =F 1, that is, that in this 


example, the value of $ = = 1. ° mh ae 
We may indeed proceed thus; but this expedient is not always 


. . dx . ' . 
sufficient when the value of ah contains more than one variable, 
> d 


nor when it contains radical quantities, even if it have but one vari- 
able. It is therefore necessary to give an easier and more general 
method. But it is first necessary to show the nature of the points 
of curve lines where this singular case occurs. lt takes place, as we 
shall soon see, at multiple points, that is, at those points where sever- 
al branches of the same curve meet. 

63 Let us conceive that SOMAWOM ( fig. 20) is a curve of Fig. 20. 
which two branches, at least, intersect at the point O. Itis evident 
that to each value of .2P, or .r, within a certain interval, there corres- 
pond several values of y,as P.M, PM’, and those beionging to the 
branches which intersect each other, become equal at the point of in- 
tersection 0. 

In like manner, 424 being the axis of ordinates, to each ordinate 
4@ within a certain extent, there must correspond several abscisses 
QN, AN’, AN’; and those belonging to the branches which inter- 
sect each other, must become equal at the point of intersection. 

If, therefore, we represent by « the value of a, and b by that of y, 
which belong to the multiple point, the equation of the curve must be 
such, that when we substitute in it a for x, we shall find for y as 
many values equal to b as there are branches passing through the 
multiple point; and when we subtitute 6 for y, we shall find a like 
number of values for x equal to a. 

It follows from this that the equation must be such that it may be 
reduced to this form, 

(c—a)™ Be (a —aym 1 y—b) F! 4 (x—a)m—2(y—b) 2 FY + (wa) —3(y=b) 8. 
tree eee ee ee tH(y—b)™ = 0. 

m indicating the degree of multiplicity of the point in question, and 
FORTE 0), Td designating quantities composed m any way of .v, y, 
and constants, or, as they are called for conciseness’ sake, functions 
of x, y, and constants. 

Indeed, it is evident, that if we make x = a, the equation which 
is then reduced to(y—b)" T= 0, will be divisible, m times, by 
y — 6, and will consequently give m times the equation y—6 = J, 
ory =. It will be the same, if we make y = 6; the equation 
which is then reduced to (.c —a)” F =0, will be divisible m times 
by «—a, and will consequently give so many times the equation 
x2 —a=0, or # =a; which cannot happen unless the equation be 
reducible to the form above. 

Letus now conceive this equation to be differentiated m times in 
succession, making also dx and dy to vary, that it may be the more 
general. If wereflect on the principle of the differentiation, we shall 
readily perceive (as will presently be demonstrated by an example), 
Ist, that the last differential equation will be the only one, in which 


60 Differential Calculus. 


there are any terms not affected by y—b, or x—a. Therefore, 
whenever there is a multiple point, the first, second, third, &c. differ- 
entials of the equation, must, when instead of x and y, are subtitut- 
ed their values a and b answering to the multiple point, be all made 
to disappear, except those, the degree of wiiose differential is marked 
by m; 2d it will also be perceived that, in this last differential 
equation, the terms affected by dd a, ddy, d® x, &c., and by all dif- 
ferentials of higher degrees, will each have as factor some power of 
x —ua, or y — b; and consequently, that these differentials will dis- 
appear at the multiple point. 

From these principles it follows, Ist. That at the multiple poimt 
“Tipe 


d 
less by the last differential equation, since all the other differential 
equations being then rendered equal to nothing, the factor of d x, as 
well as of dy, becomes each equal to zero. 2d. That as this last 
differential equation contains neither dd x. ddy, nor any higher dif- 
ferential, it might result immediately from the differentiation of the 
proposed equation, m times successively, supposing d x and dy con- 
stant. Sd. That in this last differential equation. d x and dy will 
be marked by the degree m; and, that consequently if we divide by 


we cannot have the value of expressed otherwise than by $, un- 


dy™, we shall have, by resolving the equation, m values of —, 


which will serve to find the tangents, which the different branches 
passing through the multiple point have at that point. 

To illustrate and confirm this by an example, let us take the curve 
which has for its equation 

a(y—b)? —x(xr1—v0)? = 0. 
If we differentiate this equation m times, that is, in this case, twice, 
we shall have, first, 
2ady (y— b) —da(x—a)* —2adu(x—a)=0; 
second, 
2Qaddy  (y—b) +2ady? —ddx («—a)? —2dx* (x —a) 
—2xddu(x—a)—2d x? (4 —a) —2ada? =0. 
In which we see that if we substitute a for x, and b for y, the first 
differential equation disappears, and in the second, the terms affect- 
ed by dd and dd y become nothing, so that it is reduced to 
2ady?—2adu* =0. 

But if, instead of considering da and d yas variable, in the second 
differentiation, we had considered them as constant, we should have 
had 

Qady? —2da2 (e—a)—2d22 (x—a)—22dx2 =0. 
which, on substituting a for a, is also reduced to2 ad y2—2ad x? =0, 
and gives a = -t1,a result which indicates that there are two 
tangents at the point where x =a, and y = b; this point is there- 
ydx 
dy 


fore a double point, and the value of the subtangent, becoming 


Multiple Points. G1 


then = -t b, these two tangents make an angle of 45° with the or- 
dinate. This will be confirmed by the description of the curve by 
means of its equation, which, giving 


y = b+ (x—a) ie 


shows that the curve has two branches perfectly equal and similar, 
which intersect each other at the point 0, where x = a and y = b. 
Its figure is such as represented (fig. 20). 

64. It is easy to determine by these principles whether a curve, of 
which we know the equation, have multip‘e points or not, what they 
are and where. We must differentiate the equation; put equal to 
zero the multiplier of da and that of dy. ‘These two equations will 
determine the value or values of « and y, according as there are one 
or more multiple points ; but to be assured of the existence of this 
multiple point, we must examine whether these values of «x and y 
satisfy the proposed equation. Then, to ascertain the degree of 
multiplicity of the point or points fuund, we differentiate the 
equation anew, but, for the sake of greater simplicity, considering 
dx and dy as constant. If, when the values found for « and y are 
substituted in this second differential equation, all the terms do not 
disappear. then the point found is only double. In the contrary case 
it is more than double. We proceed therefore to a third differentia- 
tion, still considering da and dy as constant: and having substitut- 
ed the values of x and y, the point will be triple if all the terms do 
not disappear, otherwise it will be at least quadruple. We continue 
to differentiate and substitute, until we arrive ata differential, of 
which all the terms are not made to disappear by the substitution of 
the values of x and y. 

For example, if it is required to find the multiple points of the 
curve, which has for its equation 


y*—anxy? +625 =0, 

we differentiate this equation, and obtain 

4y3dy—2arydy—ayrdx+3bae2dx =0. 
Making the coefficient of da and that of dy equal to zero, we have 

4y%?—2axry=0, and 3 bx? —ay? =0. 
The first of these two equations gives 
y =0, or 4y? —2ax = 0. 

The value of y = 0, being substituted in the equation 

8 ba? —ay? =0, gives $b x? =0, orz =0; 
now the preposed equation is satisfied by substituting 0 for x and y; 


therefore the curve has a multiple point corresponding to « = 0, and 
y = 0, that is, at the origin. 


| ae 
As to the value 4y2 —2az=0, or y? = aay if we substitute 


2 
= ye ar x 
itin S$ba? — ay? =0, we have $b 2? — = 0, whence we de- 


° 


= , but the first, viz. 2 = 0, gives y = 0, which 


duce 2 =0, or x = 


62 Differential Calculus. 


indicates the same point as before; from the second we deduce 


oie but these values of «and y? do not satisfy the equation 


a e 
126’ 
proposed. There is therefore no other multiple point than that found 
at the origin. 

; To ascertain its multiplicity, we differentiate a second time; we 
ave 

12y? dy?—2axdy?—2aydudy—2aydrdy + b6brdy?=0; 
of which all the terms disappear on substituting for x and y their 
yalues, zero. Therefore the point is more than double. 

We pass then to a third differentiation ; we find 
Q4ydyi —2aduxdy? —2adady?—2adady? +6bdx3 =0; 
or, substituting for 2 and y their values, zero, 

OO Ob dF Od oy = Os 

as this third differential does not disappear, the point in question isa 
triple point. 

‘To determine its tangents, we divide this equation by 66 and dy, 
dias adx 


and have Lyi — oa = 0, 


which gives 
dx da a Pe ma cs 1 
S20, and $f 0, or So mk IE 
x 


The first value, —- = 0, indicates that one of the tangents to the 


multiple point is parallel to the ordinates, that is, that one of the 
branches touches the axis of the ordinates; since the multiple point 
is also at the erigin. ‘The two values 


dx a 
Le es Je. 


show that the two other branches make with the axis of the ordinates 


each an angle of which the tangent is iB 
different sides of that axis. We may see the figure of this curve by 


resolving its equation, which gives 


x Hb Ud ES ook Sie 
yx, |" pais Vat —4be s 


2 


, and that they extend on 


taking for a and b two numbers at pleasure, and successively giving 
to x several values both positive and negative; it will be found to 
be such as is represented in figure 21. 

Finally, when we have determined a multiple point by the opera- 
tions given above, we must not always conclude that all the branches, 
which are considered as passing through that point, are visible. It 
may happen that the equation which furnishes the tangents, has imagi- 
nary roots; and then there are so many invisible branches. ‘The points 
where this happens, are sometimes detached from the course of the 
curve to which they nevertheless belong; they are then called con- 


Muluple Points. 63 


jugate points. But whether detached or not, they are not the less 
considered as containing the number of branches indicated by the 
degree of their multiplicity; the curve to which they belong is an 
individual of a more extensive family, in which all these branches 
are visible ; but they become invisible in this, because some one of the 
constant quantities which enter into the equation common to the 
whole family, becomes zero in the particular case of this individual 
curve. It is thus that in the curve, which has for its equation 


m (y-—b)? — x (x— a)? =0, 
(and of which the curve in figure 20 is a particular case, and which 
it becomes when m = a) the leaf, which this curve has, no longer 
exists when x is supposed = 0, which reduces the equation to 
m (y — b)? — x3 = 0, 

The two branches OMZ, OM’Z, which were above the point O, will 
no longer exist in this last, or at least will not be visible ; for we may 
suppose that they are still there, regarding a not as absolutely zero, 
but only as infinitely small. here will nevertheless be two tan- 
gents, in fact; but this example enables us to conceive how certain 
branches may disappear. 

65. Several useful consequences may be drawn from what has just 
been said on the subject of multiple points. 

66. 1st. When a fractional algebraical expression, into which there 
enter one or two variables, is such, that by substituting in it for each ° 
of those variables certain determinate values, it becomes 2, we shall 
obtain the value which this expression ought then to have, by differ- 
entiating separately the numerator and the denominator, as many 
times in succession as Is necessary, in order that they may not be- 
come zero at the same time; and in this differentiation we may treat 
the first differences as constants. Indeed we may always consider 


_any fractional algebraical expression ra containing two variables, for 


° | . 
instance, as being the value of de (x and y being these two varia- 
; d B 
bles); that is. we may always suppose ae mr and consequent- 


ly f4dx—Bdy=0. But since, by the supposition, 4 and B be- 


come zero at the same time, when x and y have certain values, it fol- 
: ax 
lows from what precedes that to obtain the value of Ty? We must dif- 


ferentiate this equation, considering da and dy as constant, until we 
arrive at an equation which does not disappear by the substitution 
of the values of w# and y. Now these successive difterentiations 
give 
dAdx —dBdy =0, 
dddjidx—ddBdy =0, 
d'Adx —d Bdy =0, 
whence we deduce 


—— ae OO ee Oe ees SES oo . 


64 Differential Calculus. 


that is, we must differentiate separately the numerator and denomi- 
nator as has been directed ; and the last of these equations will give 


the value of hea 
dy 


67. 2d. When an equation, which contains several variables, is 
such, that to certain values of all these variables save one, there shall 
correspond a certain number of equal values of that one. and this 
shall hold true of all the rest, we shall find these values by differen- 
tiating successively so many times less one, the equation proposed ; 
considering, in these differentiations, da.dy dz. &c. constant. if 
x, y, x, &c. are the variables; and putting equal to zero the multi- 
pliers of dx,d y, dz, &c., those of d.r?,dady.dzdy, &c.. and so 
throughout. If all these equations correspond with each other and 
with the equation proposed, the values of a, y, x, &c., which they 
give, will be the values required. 

68. It may be observed with regard to multiple points, that since 
the values of x and y. found by putting equal to zero the coefficient 
of dx and dy, must satisfy the nite equation proposed, we cannot, 
unless the equation is given in finite terms, or nay be reduced to 
them by the methods of the integrai calculus, ascertain, by calcula- 
tion alone, the existence of those points. 


Of the visible and invisible points of inflexion.t 


69. It sometimes happens, that a branch of a curve, after having 
been concave towards a certain point, becomes afterwards, in its pro- 
gress, convex towards it (without discontinuing its course). Such is 
the curve represented in figure 22. The point O, where this change 
takes place. is called the point of infleaion. 

To determine these points it must be observed that the tangent to 
the point O must be a common tangent of the branch OB and the 
branch Ob, which meet at the point O; if, therefore. on each side of 
0, we take two arcs, equal or unequal, but infinitely small, the tan- 
gent must be the prolongation of each arc. so that the two little arcs 
must be in the same straight line. 

‘This being laid down, we draw the ordinates MP, OQ, mp. and 
call 4P, x, and Py, We then have Mr—=d.x, Or =d y, and 
supposing G.if and Om to differ infinitely little from each other, we 
have 

Or'=d.(wt+da\=da+dda, and mr =dy+ddy; 
since then AZO and Om are in a‘straight line, the triangles Mr O 
and Or' m are similar: and if we suppose, as we are at liberty to do, 
that the ares AZO and Om are equal, these triangles will be also 
equal, and we shall have Mr = Or’, and Or =m 71’; that is, 

dx=dx+dda, and dy=dy+ddy, 
therefore 


ddx=0, andddy =0. 


t These are sometimes called points of contrary flexure. 


Points of Inflexion. 6d 


In order, therefore, to determine. the simple points of inflexion, 
we must differentiate twice successively the equation of the curve, 
and consider, in the second differential equation, ddx=0, and. 
ddy =0. Now it is evident that, in that case, this second equation 
is the same as if we had differentiated, considering dx and d y con- 
stant. iftherefore, from the first differential equation we deduce the 
value of da or dy and substitute it in the second, we shall have an 
equation, which, being divided by dy? or d x, will contain only «x, 
y and constants, and which, being compared with the equation of the 
curve, will give the values of x and y, which correspond to the sim- 
ple point of inflexion. . 

Let us take, as an example, the curve which has for its equation 

0S mm b y? = a>, 
We shall have 8x2%*duax—2bydy =0, 
differentiating again, treating d x and dy as constant, we have 


the first of these gives 


3.2702 
O41 oe ortigead 
substituting this value in the second, we have 
18b «+d x? 
6 xd x? may 4 = (), 
or 4by? —3xu° =0. 


3 
From this we deduce y2 = —*—, which, being substituted in the 


equation of the curve, gives 
x3 —3 73 = a3, or x? = 445, 


therefore ise Of 4, 
and consequently a 
aia (9 aera i 3a 
ae prem rsaet ome 


These are the values which determine the point of inflexion. 
Let us take, as a second example, the curve which has for its equa- 


BS 
tion y=at+(x—a), 
—2. 
we have dy =3(4—a) *dx, 
differentiating anew, considering dy and d « as constant, we find 
Per Gamat 
— ~ (2 — dt) 5 d x2 Be som 2 ada SOs 
(« —a)s 


therefore d x = 0; now the first differential equation 


2 
dy =3(x#—a) dz, 
becomes 


66 Differential Calculus. 


2 
but since dx = 0, we have (v—a)'dy =0, which gives either 
dy = 0, or (w—a)> =0; but as itis not possible that we should 


2 
have, at the same time dw and dy =9, it follows that (vw —a)> = 0 
is the true solution, which gives « = a, and consequently y =a. 

70. We may here observe, Ist, that as we findd x = 0, the tan- 
gent at the point of inflexion of this curve is parallel to the ordi- 
nates. 

2d. If the curve should have several points of inflexion, the final 
equation would give several values of x, that is, it would exceed the 
first degree. This takes place in curves which have a serpentine 
course, as in figure 23. 

71. If we conceive that the two points of inflexion O and O’ 

Fig. 23. (fig. 23) approach each other continually, and are at last infinitely 
near each other; then, if we represent, as above, the two infinitely small 
arcs O.Mf and Om, and the two other infinitely small arcs O’.M’ and 
O’m', on each side of the points of inflexion O and O’, the two sides 
O m and M’0' will be, or may be supposed to be, one on the other ; and 
since, at the point of inflexion, MO is in a straight line with Om, 
and Mé’O’ with O'm’, there will then be three small consecutive arcs 
in a straight line. 

Fig.24, This being laid down, let Mm, mm’, m’m’’ (fig. 24) be these 
three infinitely small arcs. Het fall the ordinates MP, mp, m’ p’, 
m” p’, and draw the lines Mr, mr’, m’r’’, parallel to AP. Cail 
AP, x, and PM, y. We have 

Mr=darm=dy, mr =dx+ddxa, r'm'=dy+ddy, 

mr’ =dx+t+ddxtdsaz, r’m’ =dy +ddy +d y. 
Now the three triangles Mrm, air’ m’‘, m'r"m” are similar, since 
the sides fm, mm’, m’m’ are in a straight line; if therefore we 
suppose these sides equal, which we are at liberty to do, the triangles 
will be equal. We shall have then 
dx=dx+ddzdy=dy+ddy,datddsr=da«e+ddz +d3z, 
dyt+tddy=dy+ddy+d'3¥y; 
that is, 
ddx=0, ddy =0, d' a= 0, d?y¥=0. 

If, therefore, we differentiate the equation of the curve three times 
successively, considering as variable dz, dy, dda, ddy, and af- 
terwards put 0 for each of the quantities dd a, dd y, d3 x, d3y, each of 
the three equations resulting from those differentiations, will hold true. 
Now it is evident that they then become the same as if we had dif- 
ferentiated three times in succession, supposing d x and dy constant. 

By a similar course of reasoning it may be shown that if three con- 
secutive points of intlexion come to unite, there will be, at the point 
of union, four elements in a straight line, and it may be proved in 
the same way, that if we differentiate the equation four times in suc- 
cession, supposing dx and dy constant, the four resulting equations 
will hold true ; and so of others. 

_ Therefore, if, from the first differential equation, we deduce the 
value of dx, and substitute it in all the others, we shall have, after 


Points of Inflexion. 67 


dividing the second by d y?, the third by d y5, and so forth, so many 
equations, which must hold true conjointly with the proposed equa- 
tion, in order that there may be one, two, three, &c. inflexions unit- 
ed__ If, therefore, from two of these equations we deduce the values 
of «and y, these values, substituted in the other equations, must sat- 
isfy them. 

When there are only two inflexions united, the inflexion is invisi- 
ble, it becomes visible when there are three ; in general, the inflex- 
ion is visible or invisible, according as the number of inflexions united 
is odd or even.. Therefore if H = 0, represent generally the equa- 
tion of a curve, it will be necessary, in order that there should be 
m points of inflexion united, that, differentiating H, m + 1 times, on 
the supposition that dz and dy are constant, all the differentials 
ddx,ddy,d x, ds y, &c. to that of the degree m + 1, inclusively, 
should be zero; and the inflexion will be visible or invisible accord- 
ing as m is odd or even. 

72. Hitherto the ordinates have been supposed parallel. If they 
issue from a fixed point, the following is the method of determining 
the points of inflexion. Imagining the two infinitely small consecu- 
tive arcs, which, at the point of inflexion, must be in a straight line, 
we draw the ordinates (fig. 25) CM. Cm. Cm’, and describe the pig, 95, 
arcs Mr. mr’. which may be regarded as perpendicular to Cm and Cm’. 
This done, it is easy to see that the angle r’ m m’ differs from the an- 
gle r Mm by the angle m Cr’. For we have C m im’ + rm M = 180°, 
or Cmr' + r' mn’ + 90° —r Mm = 180° ; 
therefore yr mm —rMm =90°—Cmr' ;sx 
but by the triangle Cr’ m right-angled at r’, we have 

90° = Cmr'+m Cr’; 
therefore rom —rMm =m Cr’. 

If we draw the line mn, making the angle m’mn =m Cr’, the 
angle nmr’ will be equal to m Myr, and consequently the triangle 
tmr’ will be similar to the triangle m Mr. Calling OM, y, and 
Mr, d «x, we have 

mr =dy, mr =dx+ddxa, and mr =dy+ddy; 
call Mm, ds, mm’ will be =ds-+dds. Describe from the 
point m with radius = mm’, the arc m’n: the sectors C mr’ and 
and nm m/ will be similar, and will give 
Cm:mr':: mm’: mn; 

that is, ytdy:dx+ddx::ds+dds: wn, 
which, omitting the quantities which may be neglected, gives 

: dsdx 

y 

but the triangle m’tn, which is similar to tr’ m, will be so likewise 
to mr M; we have, therefore, Mr : Mim :: m’n : m’'t; that is, 


2 
dat de ee as 


ds? 
therefore wt=dyt+tddy— aie 


68 Differential Calculus. 


Now the similar triangles Mr m and mr’t give 
Mre:rm::mr: r’t, 


or daidy::de4ddeidy4ddy——; 
therefore 
dedg + brad ye Coen iy 4 aidan 
2 
or daly ay der ee Ee 


this is the formula for finding the simple points of inflexion, whem 
the ordinates issue from a fixed point; it becomes that for the paral- 
fel ordinates, when we suppose the point C at an infinite distance, 


dad s? 
and dx and dy constant; for then the term ~— must be reject- 


ed, as y is infinite. In the application of this fran it will always 
be more simple to suppose d x constant, which reduces it to 
2 

and care must be taken to treat d x as constant in the differentiation of 
the equation of the curve : but as the lines d x are arcs described with 
a variable radius, while, if the ordinates issue from a fixed point, they 
are referred to arcs described with a constant radius CO, as has been 
observed (39), we must be careful to substitute for d x its value, which 
will always be easily found, by observing that the sectors CSs and 
CMr are similar. It is not necessary to give the formulas for the 


other points of inflexion, on the same supposition, although it would 
not be difficult. 


Observations on the Maxima and Minima. 


73. Let F be a function of one or more variables, and let it be 
susceptible of a maximum or minimum. If we imagine it to repre- 
sent the ordinate or absciss x of a curve, that is, if we suppose x = F, 
it follows from what has been said on the subject of the points of in- 
flexion, that if it be true that d F or dx becomes zero, whenever 
there is a maximum or a minimum, the converse is not always true 5 
for we have seen a case (69) where dx was found =0 at the point 
of inflexion, that is, the tangent became parallel to the ordinates ; 
but it is evident that neither the ordinate nor the absciss is then 
either a maximum or minimum. What then must be done to ascer- 
tain the existence of a maximum or minimum? We must differen- 
tiate the quantity several times in succession, considering the first 
differentials of each variable as constant; and, if the values’ which 
the variables have at the point of the maximum or minimum 
sought, cause dF, and dd¥F to disappear, but not d° F, there is 
no maximum in the curve which has for its equation «=; but 
there is a visible point of inflexion; so that the quantity F has no 
maximum or minimum. But if d3F disappears, and not d* F, 


Cusps. 69 


there will be a mazimum or minimum. In general, there will be a 
maximum or minimum, if the last differential made to disappear is 
of a degree marked by an odd number. 


Of Cusps of different species, and of the different sorts of contact of 
the branches of the same curve. 


74. When two branches of a curve come in contact, they may either 
have their convexities opposed to each other as in figure 26, or have 
their concavities opposed, and thus one embrace the other, as in figure 
27, and in both cases it may either happen that they continue their 
course on each side of the point of contact, or that they stop sud- 
denly at that point, as we see in figures 28 and 29. In this last case, 
the point of contact is called a cusp ; that represented in figure 28 
is called a simple cusp or cusp of the first species; that in figure 29 
is called a beaked cusp or cusp of the second species. If more than two 
branches unite, their different varieties may be found at once at the 
same point, and there may also be found an infinite number of others ; 
for example, the branches may, at the point of contact, undergo one 
or more inflexions. It may happen that an inflexion and a cusp oc- 
cur united. so as to seem to form only a single cusp In figure 30, 
if the branch EBD, which forms at #, with the branch HC, a cusp 
of the first species, had a point of infiection at B. and if the point of 
inflexion B should be infinitely near the point #. we should only be 
presented with the appearance of a cusp of the second species. 
These varieties may be infinite, especially if we consider that sever- 
al branches may touch at once. We shall not undertake to give the 
character of each; we shall only observe, that whenever several 
branches of a curve touch each other, it may be ascertained by the 
following facts. 1st. This point being multiple must have the con- 
ditions enumerated in art. 63. 2d. The equation which serves to de- 
termine the tangents of the multiple points, must then have as many 
equal roots as there are branches which touch, since there must be so 
many tangents united. Thus. for cusps of the first species, there 
ought to be the conditions common to double points, and the equation 


which gives the tangents thereof, or which gives 7 ought to have 
two equal roots. As the consideration of these points is not suffi- 
ciently useful to engage us in the details which their investigation 
would require, the subject will be no farther pursued. 


On the Radii of Curvature and the development or evolute. 


75. If upon each of the points JZ, m, m’, &c. of any curve line 
( fig. 31), we conceive the perpendiculars MN’, mn, m’n', &c. to be 
raised ; the consecutive intersections JV, n,n’, will form a curve line, 
to which has been given the name of evolute, because if we consid- 
er it as enveloped by a thread ABM, which touches it in its origin B, 
then, upon unwinding the thread, the extremity 2 traces the curve 2M. 


Fig. 31. 


Fig. 32. 


Fig. 33. 


7 Differential Calculus. 


In fact, in the development of Vn, for example. considering Vn as 
a sinall straight line, the thread MN n describes, about the point n, 
as a centre, the arc Wm, to which it is necessarily perpendicular, 
since the radius of a circle is perpendicular to its circumference. 

76. The curve AM being given, if we wish, for any point M, to 
determine the value of 4m, which is called the radius of curvature, 
we observe that Zn is determined by the concourse of two perpen- 
diculars infinitely near each other, MV and mn. We therefore im- 
agine (fig. 32) two consecutive arcs Mm, mm’, infinitely near. and 
differing infinitely little from each other. which may be considered as 
two straight lines; we also imagine JZ’ perpendicular to Mm at 
M, and mW perpendicular to mm’ atm. ‘Then, in the triangle 
NVM m, right-angled at Af, we shall have 

1:sinUNVm::mVor MM: Mm, 


or, because the angle JV’ m is infinitely small, 


“1: MNVm:: MV: Mm; therefore VM — ot 


MN’ m ” 

but, if we produce Mm, we have Vmu =VMm-+MNm; be- 
cause MWmnu=Vmm-+ ni mu= Nilm + JEN m, and taking 
away Vim’ = JV im, there remains m’ mu = MN m; therefore 


MN = Sooidnaa 
m! in u 
If we draw Mr and m7?‘ parallel to 4P, it is easy to see that the 
angle wm 1’ being equal to m Mr, the angle wm m/is the quantity by 
which the angle m Afr is diminished, or the differential of the angle 
mMr,which must be taken negatively here, where the curve is concave, 
and where it is convex, must have the sign plus; we have there- 


M Mm 
fore MN = sd(rMm)- 


sion ford (r Mm). Now the tangent of r Mm = ae and 


We have therefore to find the expres- 


cosr Mm = she 
ds 
ds designating the arc Mm; but we have seen (23), that z being any 


arc, we had dz = cos z d (tang z); therefore 
oe bx? dy \. 
d(r Mm) = PRS d (5); 
d 


ds s$ 
ds® (3 ee aa 
This is the formula for finding the radius of curvature when the or- 
dinates are parallel 
77. If the ordimates are supposed to issue from a fixed point 
( fig 33), we have, as above, and for the same reasons, 
s MN = AMZ m 


mm u 


whence, 


« 


sa 


Radu of Curvature. 71 


but m/ mu is no longer = == d(r Mm), but the value of this angle is 
found as follows. 
Describing the arcs Mr, m7’, we have 
ma’ mu=s=rmu—rmm’; ; 

but it was shown (72) that 1’ mw differs from r.Mm by the angular 
quantity m Cr’ or MC r, for the two last quantities differ from each 
other only by a quantity infinitely small compared with them ; there- 
fore r’'mu = MCr + rMm; therefore, since 


mmus=rnu—r mm, 
we have 
m’ mu = MCr+rMm—r' mm = MCr—d(r Mm). 
Now, calling Mr, dx, we have 
| 1: sin M@Cr or: MCr:: CM: Mr; 


that is, 1: MCr:: y : dx; therefore MCr = = and since 


2 
EGAN, ae oH), 


« ds? dx 
Coles aL? dy 
/ e 
we have m’ mu = ard d a, )3 


and if the curve were convex, we should find, in like manner, 
dx. da> dy 
/ nade con Lea Sait 
therefore 


3 
MN = sa I, 


dx aba ad aa du 
arr ret q) dstdazyded (Se 
78. To give an application of these formulas, let us suppose that 
the curve 4M (fig. 32) is a circle, having for its equation Fig, 32. 
y? = 2axr—x*, 
we shall have y = ./2axz—vx2 ; therefore 
ae dix—adx 
A / Dogan, 
and consequently, 


ds= /d x2 +dy2 = 
cally as Dd 
and OM, = pladeie tas ; therefore d (34 — peer Oc AY, 
dx V2ax— x2 (2a x —a2)2 
the formula which belongs to the present case, in which the curve is 


yA aaa aS 
concave and the ordinates parallel, is oR er 


which is changed into 


72 Differential Calculus. 


az das 
‘ Cae ys 
a2 d «3 
x Qax—x? yt 
that is, the radius of curvature is always of the same magnitude and 
equal to the radius of the circle; so that the evolute is reduced to a 
point, which is the centre; and this agrees with what we know of the 


circle. 
Let us take, as a second example, the parabola which has for its 


, alts a4 
equation y2 =a, ory = /ax = a? x, we shall have 


by 
dy=30 x *da, 


therefore 
ds=/da? +dy? =4/dx* +laa— ld x? mde 14 chet 
4a, 
Arta —1 —_——— d tthe tds 
=adx spre ae Pde fiafa,and 4 = 30x 23 
‘ dy 1 z a 
therefore d Wey ee dx; 


d 53 


the formula ne 
—adxid Ce. 
dx 


becomes then 


re) 3 3 pelle ey 
2o 2dx3(4u-+a)?  4(42+ a)? aoe fete 


SS eee ae Ura py 
da? Xiata 2dx az 


79, The radii of curvature serve to measure the curvature of a 


curve at each point. Since, in the development of the element nn’ 


Fig. 31. 


of the curve BV ( fig. 31), the thread traces the small arc mm’, this 
arc has the same curvature as the circle which has for its radius the 
line mn. ‘Thus when we have the expression for the radius of cur- 
vature, we know, for each point, the radius of the circle which has 
the same curvature as the curve has at that point. And as the cur- 
vature of a circle is greater in proportion as its radius is less, that 
is, as the curvatures of circles are in the inverse ratio of their radii, 
it will be easy to compare the curvature of a curve at any point with 
the curvature of the same or another curve at another point. Thus, 
if we wish to compare the curvature of the parabola at its origin, 
with that of the same curve when the ordinate passes through the 
focus. we observe that at the origin « = 0, and that the absciss cor- 
responding to the focus is }.a (4p.172). Putting therefore, succes- 
sively, for the expression of the radius of curvature, x =0, and 
a ==1La,we have $a, and a./@, the radius of curvature is there- 
fore 4a at the origin, and a ,./@ at the extremity of the ordinate 
which passes through the focus, therefore the curvature at the first 
of these points is to the curvature at the second, as 


OW Busi Gg, OF 2308 4/ 2) 6 Le 


y) 


Radius of Curvature. q 


Since the radius of curvature MN’ is nothing but the thread which 
is supposed to have enveloped or been wrapped about the curve BN, 
it follows that it is equal in length to the arc BN, plus the part AB, 
by which the thread exceeded the curve when the development be- 
gan, that is, plus the radius of curvature at the origin 4. Therefore 
the curve BN’ is rectifiable, that is, we may assign the length of each 
of its arcs BN. 


Remark. 


80. The points of inflexion were determined on the supposition 
(69) that the two elements of the curve, near the point of inflexion, 
were in a straight line. From this supposition it seems to follow, 
that at the point of inflexion the radius of curvature must always be 
infinite, because the two perpendiculars upon the two consecutive 
sides must be parallel. ‘There are, nevertheless, several curves 
which. at the point of inflexion, have the radius of curvature equal 
to zero; the parabola, for example, which has for its equation 

= x8, 

But it must be observed that nothing in the supposition that was 
made, determines of what magnitude those two consecutive elements 
are. Now if they be each reduced toa point, they will be never- 
theless in the same straight line, and the two perpendiculars falling 
one on the other, will meet at the very point from which they issue. 
And this is what happens in those curves where the radius of curva- 
ture is zero at the point of inflexion For the curvature then being 
infinite, the two consecutive elements are each confounded with the 
tangent infinitely less than in any other case. and must consequent- 
ly be considered as two points united. The two elements may there- 
fore be in a straight line, without the radius of curvature being infi- 
nite ; but we see from this, that the radius of curvature is, at the 
point of inflexion, either infinite or nothing. 


10 


ELEMENTS 


OF THE 


INTEGRAL CALCULUS. 


Explanations. 


81. The method known by the name of the Integral Calcu- 
lus is the reverse of the Differential Calculus. It has for its ob- 
ject to ascend from differential quantities to the functions from 
which they are derived. 

There is no variable quantity expressed algebraically of which 
we cannot find the differential; but there are many differential 
quantities} which we cannot integrate; some, indeed, because 
they could not have resulted from any differentiation ; such are 
the quantities vdy, cdy—ady, &c., others, because means 
have not yet been discovered of integrating them; and among 
these last are some of which we may despair of ever finding the 
integral. 

As, however, great use may be made of those which we know 
how to integrate, we shall endeavor to show the methods, and 
shall afterwards show what is to be done with reference to those 
which refuse to be integrated. We shall begin by explaining cer- 
tain modes of expression which will hereafter be made use of. 

We call a function of one or more quantities, any expression, in 
which those quantities enter in any way whatever, whether mix- 
ed or not with other quantities which are considered as having 
determinate and invariable values, while the quantities of the 
function may have all possible values. ‘Thus, in a function, we 
consider only the quantities which are supposed variable, without 


t By differential quantity is meant here, not only such as results 
from a differentiation, but in general, every quantity affected by the 
differentials d wv, dy, &c. of one or more variables. 


Simple Differentials. 15 


any regard to the constants which may occur in it. For exam- 


ple, 7, a+6 x2, Wire carer &c. are functions of x. 

By algebraical quantities is meant those, of which the exact 
value may be assigned, by executing a determinate number of al- 
gebraical or arithmetical operations, other than those which de- 
pend on logarithms. On the contrary, we call non-algebraical or 
transcendental, those quantities, for which we can assign only prox- 
imate values, or by means of approximations ; logarithms are of 
this kind, and an infinite number of others. 

To indicate the integral of a differential, the letter ff is written 


before this quantity ; this letter is equivalent to the words sum of, 
because, to integrate, or take the integral, is nothing but to sum up 
all the infinitely small increments which the quantity must have 
received, to arrive at a determinate, finite state. 


Of the Differentials with a single variable, which have an alge- 
braical integral ; and first, of simple differentials. 


82. Fundamental rule. To integrate a simple differential, we 
must, Ist, increase the exponent of the variable by unity ; 2d, di- 
vide by this exponent thus increased, and by the differential of the 
variable; that is, divide by this new exponent multiplied by the 
differential of the varrable. 

The reason of this rule is evident from the principle of differen- 
tiation (10). As the object is to find the quantity which must have 
been differentiated, it is evident, that we must make use of ope- 
rations the reverse of those employed for differentiating a quan- 
tity. This is made more clear by examples of the application of 


the rule. 


Qxu11d x 
aa 1 eae in eee Se — m2 e 
jfreder= fra da eas oo ; 
SRR 
5 Sao s. 
We see, now, et A is actually 2ada” 3 and 
‘ oy male 
a(> =crdux. 
In like manner, 
; 5 
= fee as ax 5 
OE os a bal i's 
fas BPE) dx z Sa Di) 5 


76 Integral Calculus. 


ada ; wie aw 3t+lda ax* a 
= = far ede: = ——___. = —_ = — — 


3 
Xx 


(—3 +1)dx” —2 22° 
In general, m being an exponent, positive, negative, integer, or 
fractional, we have 


fl amt dx a ams 
fia “aged dx omen yf eee es aA a 
(m+ 1)dx m+ 1 
We have no need of this rule to find the integral of dw or 
adx, which, we see at once, must be z for the first and aa for 
the second. But, if we wish to apply the rule to them, we ob- 
serve that the exponent of w in these differentials is zero, and 
that they are the same thing as 7° dv and a x° da, of which the 
xT dx a xt da 
—————— and -— ——;—_;, 0 
O+i)dz (+ 1)dex 


— 


integral, agreeably to the rule, is raz 


and a 2. 


There is only one case which eludes this fundamental rule; it 
is that in which the exponent m has the value — 1; for then the 
Ue aa Qa B 
oe eT F ahr Aah quantity unassignable 
because infinite ; for, if we conceive the denominator, instead of 
being zero, to be an infinitely small quantity, it must evidently be 
contained an infinite number of times in the finite quantity a, and 
consequently the fraction must be infinite. We shall explain, 
hereafter, the reason why the calculus gives in this case an infi- 


nite quantity ; meanwhile we observe that the proposed differen- 


cf ,is the differ- 
ential of a logarithm. It is the differential of al x or of 1 2%, as 
may be easily seen by differentiating (26). 


If there were a radical in the simple differential, we should 
substitute, for the radical, a fractional exponent. Thus, 


3 2 5 
fadan/x =faaxtdo=2ac%. 


integral becomes 


ad 


tial, a x" d x, which in this case is aa—" d x, or 


4 


Remark. 


33. We have seen, that when, in the quantities to be. differen- 
tiated, there occur terms wholly constant, these terms do not ap- 
pear in the differential. When, therefore, we go back to the in- 
tegral, we must take care to add a constant quantity to the result 
of the integration. ‘I’his quantity will always have an indetermi- 


Complex Differentials. 77 


nate value, as we have no other object than to find the integral, 
that is. to find a quantity, such that, by differentiating it, we re- 
produce the differential proposed 5 indeed, 

aan rr axmtt 

ee ne eats 
C being any constant quantity, have equally for their differential 
the quantity az” da, whatever value is given to C. But when 
the integration is performed with a view of satisfying a given 
question, then this constant quantity has a value, determined by 
the state of the question. ‘This will be shown hereafter; but, 
in future, care will be taken to add a constant quantity to the re- 
sult of each integration; and that it may be known as such, it 
will be always designated by the same letter C. 


Of Complex Differentials whose integration depends on the 


fundamental rule. 


84. 1st. We may integrate by the preceding rule, any quantity, 
in which there occur no powers of complex quantities, and no 
complex divisors, except such as are constant quantities. 

Thus the entire integral of 

bx? ds 
08! dah pe 


being the sum of the integrals of each of the terms, will be 


In like manner, 


f(aridat noe 


- ) =f (408 de+ba* da) 


a .x* b 
rm Ramee Sigiiak C. 


85. 2d. Even should there occur powers of complex quanti- 
ties, they may still be integrated by the fundamental rule, provid- 
ed they be not found in the denominator, and provided, also, 
their exponent be a whole positive number. For example, 
(a +6 x?) dx may be integrated by the preceding rule, by ac- 
tually raismg a+ 6a? to the third power, which would give 
(Algebra 141) a3 +5@2b 2? +306? a+ +465 x8, and conse- 
quently 


78 Integral Calculus. 


f (a46 w?)8dae=ffaidx4+3a%be2dr43ab2 14d x+b3 a d at 
= fas dx+ {[3a? b x* dx+f3ab? rida + ['b3 xs da 

BA, 0 ee Bi 0 eee = 

Aegis Sapa Fe 

86. As there is no complex quantity 1 to a power indicat- 
ed by a positive whole number, which may not, by the preceding 
rule (lg. 141), be thus reduced to a finite series of simple 
quantities, we may always integrate any complex quantity which 
does not contain any other complex parts than powers whose ex- 
ponent is a positive whole number. ‘Thus, if we had to integrate 

guidxe(atba?)?*+a%vti'dx(c+ex?+fu3)4, 
we should develope, by the rule already cited, the value of 
(a +6 x?)?, and multiply each term of the result by gv° da; 
we should in like manner develope the value of (c +ea* + fx3)4, 
and multiply each term of the result by a? 27 dx; we should 
then have only to integrate a series of simple quantities, which 
falls under the fundamental rule. 

87. We must however except the case, in which some one of 
the exponents being negative, it should happen, after the develop- 
ment and multiplication, that the exponent of the variable in 
some of the terms became = — 1; but we should then integrate 
by logarithms. For example, if we had 


hath -~(a-4+b x?)?, or aa-S$dau (a+b x7)? ; 


= ara + 


we should ae it into 
ax—*dx(a2?+2ab x? +6? x4), 
which becomes 
as asda+2a2ba-da+tabrada, 
of which the two terms one da and ab? «1 dx have, for their 
am 1% a2 a h2 a 


oT ; but the term 2 a? 6 a—'d x, which 


integral, 
: 52, adnx . Beye. eg : 

is the same as 2 a? 6 Ta ds (27) the logarithmic differential of 
2a?bl «3 so that 


fac duatbh x?)?=— 


88. 3d. If the differential cae proposed even contains a 
complex quantity, raised to any power (whether its exponent be 
positive or negative, whole or fractional), we may still integrate. 


ee ols v2 


ut 2 4.2076 los. a + 


Binomial Differentials. : 79 


if the whole of the terms multiplied by this complex quantity, 
taken together, be the differential of the complex quantity con- 
sidered without its total exponent; or if it be this differential 
multiplied or divided by a constant number. We have only, in 
that case, to consider the complex quantity in question as a single 
variable, and apply, word for word, the fundamental rule. For 
example, g.dx(a+6.)? falls under this case, because g d x 


is the differential of a + 6 a2, multiplied by = which is a con- 


stant quantity ; so that, to integrate it, we write 


_ gdx(atbarth 
Peas Ot BO Ta bay 
Su) NaS CR ese 


y__ Slat baler} 
Gaiide GS Gee 


For, if we differentiate this quantity, we find again 
gdx(a+bx). 
In like manner, if we examine the differential 
ES (a? dx+2axdx) (ax 402)°?, 
we shall find that it is integrable, because a?@dx-+2aad is 
the differential of a+ «3, multiplied by a constant quantity a. 
By applying the rule, therefore, we have 


f(@ dx+2axdx) (a ape?) 2 
; 1 
_(adx+2 axd x) (ax+a?)? 
aa diadx +2xdx) 
The only case to be excepted is that, in which the exponent of 


the constant quantity should be —1; we should then integrate 
by logarithms, as will be seen hereafter. 


4+C=2a(au4a2)7 40. 


Of Binomial Differentials which may be integrated algebraically. 


89. We understand by a binomial differential one, in which the 
complex quantity, however complex, is some power of a bino- 


mial. Thus gat dx (a+6 a2)8 is a binomial differential. The 
same may be said of ga™dx(a+62")?, which may represent 
any binomial differential, because, by g, a,b, m, n, p, we under- 
stand any imaginable numbers, whether positive or negative. 
There are no means of integrating generally every binomial 


80 Integral Calculus. 


differential. But it is apparent, from what has already been said, 
that we can integrate a binomial differential ga” d xv (a +ba”)?, 
in the two following cases. 

Ist. When p ws any positive whole number, whatever may be the 
exponents m and n (85), with the exception of the case mentioned 
im art. 87. 

2d. When m, the exponent of x out of the binomial, is less by 
unity than n, the exponent of x in the binomial ; that is, we may 
integrate generally, g a"—1d x(a +6 a” )?, whatever value n and 
p may have, except the case in which p =— 1. In fact,g a*7~1d x 


is the differential of a + 6a”, multiplied by =, that is, by a con- 


stant quantity ; we fall then upon the case mentioned in art. 89 ; 
and may consequently integrate by the general rule, considering 
a-+6x" asa single quantity. 

Besides these cases there are two others, which may be com- 
prehended in one, and which include the preceding ; they are the 
following. | | 

90. Ist. We may integrate any binomial differential, in which 
the exponent of x out of the binomial, being increased by unity, 
may be exactly divided by the exponent of x in the binomial, and 
give for a quotient a positive whole number. The process to be 
followed in this case to integrate, and also to show that the prin- 
ciple is general, consists in making the binomial quantity, without 
its total exponent, equal to a single variable, and expressing the 
proposed differential by means of this single variable and con- 
stants ; which may always be done by proceeding as in the fol- 
lowing examples. 


Let it first be proposed to integrate g x*° dx(a+b a), We 
see that this differential may be integrated, because the exponent 
of x out of the binomial, viz. 3, being increased by unity, gives 
4, which, divided by 2, the exponent of «x in the binomial, gives 
for a quotient the positive whole number 2. 

We make, therefore,a--6a”?=2. From this equation we 


Z— it 
- We observe 


deduce the value of «”*, which is x? = 


that x? d x, which precedes the binomial quantity, results, except- 
ing its constant multiplier, from the differentiation of x4, the square 


Binomial Differentials. 81 


: : Zee 
of 22; we therefore square the equation 7? = ; 


on and find 


Z—a\? : Bes 
x= (- ; ‘) ; differentiating, we have 


Sey. (PN ae 
Aw da =2( ; ,? 


and consequently 


x8 d v= (=) _ (%—a)dz 


2 b? 
Substituting for v3 dx and (a'+ 6 oa their values in terms of z, 


in the expression g v3 dx (a+6 x2), we find 


4 
g.(z—a)dz eet aoe de gaat dx 
om 2b? 2, b% 
Then 
¥ 4 4 
4 $b ty arr) 
a rainy Ate xz [gaxrdz 
fs dx (a-+6 x?) = Kea f[E- 
gut? gaz t F, 
= eka a —(e¢4F12e t 
hgiet Beye) Shey 
ects Obs 442 441 + 


A 
Ure, all SOT eA Tire +C; 
2 b2 re ea ; 


substituting then for z, its value a+ 6.x?, we have 


5 5 
3 $ — 2 $+ EXya » eae 
fer dx(a+bx2)\> = = 5, (a-+-bx?) ‘So (athe ) sep te. 
91. We proceed i in asimilar manner in every other case sub- 
ject to the same conditions. Let us take, for example, 


guidua(a+ bas) 78, 
which must be integrable, since the exponent 8 increased by 1, 
that is, 9, being divided by 3, the exponent of @ in the binomial, 
gives a positive whole number. We therefore make a + 603 =2 ; 


and find 23 = ~_ 


a . ° : 
, and as v* d a, which precedes the binomial, 


is, excepting its constant factor, the differential of x°, in order to 


. Z— Ot —a\3 
have w° we cube the equation «3 = 53 We have 7° = ( ; ) ; 


differentiating in order to obtain «* d x, we have 
a 


82 Integral Calculus. 


Q92ridxr=3. =). F 


and, consequently, 
8de=(2—* : es 
x* dx ( ; ) fem 


The differential ¢ x? dx (a+ 6 8) 3 will be therefore changed 
; Z—aA\2 dz 2. ; 
into g@. C7") fie 7 =, by expanding 


Q—2 eee —2 
ae to BF az ee dz pee am eee, 2 3 $e 2 weaz 
PU SY 3 b3 3 3 
of ses the integral is 


Q—2 ; Piste: § 
Bie: Saal fly Nasfnd Soca aang ey Scalia Dek 
3 63 (6 — 2) 3 b3(2— 2 3 6% (1 — 2) 
s 
rie 


g 1—2 salted 
© 
3b3 * (SS mc a Eee a ae 
og 1—2 /3 ~2 ba: Zz 
& ig) 2 
= sis CS Re aS 
or finally, substituting for z its value a +0 x3, we find 


fee’ de(a4bo)? 
2 1—2(3 6 Q : 
= aye (@tbe%) *) > (a+b0°)? —2 (abe) a4Sa? } + C. 


: ; a l—2 4 
which, separating the common multiplier z ~%,is reduced to 


Such is the method to be pursued whenever the exponent of x 
out of the binomial, being increased by unity, and divided by the 
exponent of win the binomial, shall give as quotient a positive 
whole number. ‘ 

92. 2d. Although a binomial differential quantity may not fall 
under the case of which we have just been speaking, it often 
happens that it may be reduced to it by means of a very simple 
artifice, which consists in rendering negative the exponent of & in 
the binomial, when it is positive, and rendering it positive when it 
is negative. In order to this, we must divide the two terms of the 
binomial by the power of a in the binomial, and multiply the 
quantity out of the binomial by this same power of w raised to the 
power indicated by the total exponent of the binomial. For ex- 
ample, in order to render negative the exponent 2 of a, in the 
binomial gardae(a+bies)*, 


Binomial Differentials. 83 


we divide a + 6 x* by «?, which gives 9 a4 dx (= +6 >; or 


gvtdux(aa-?+6)$ ; but, as the quantity 2, by which we have 
divided, is considered as raised to the fifth power, since it is un- 
der the total exponent 5 of the binomial, we must, in compensa- 
tion, multiply the quantity out of the binomial by (x?)*, that is, by 
21°, which gives gvi*dax(aa-?+6)5. 

By this preparation, many binomial differentials which would 
not otherwise be comprehended in the preceding case, will be re- 
duced to it. For example, if it were required to integrate 

ard x 
(a2 + we?) 
we perceive that the exponent of x out of the binomial, that is 
to say, 0, being increased by 1, which makes it 1, cannot be exact- 
ly divided by 2, the exponent of x in the binomial ; but we must 
not thence conclude that the proposed quantity is not integrable ; 
for, if we render negative the power of g in the binomial, by 


vo HE Nae RF 402)? ; 


oa: _ —3 : : 
writing a? (a?) ?da(a2.« +1) 2, which is reduced to 


a? ord ea? 27 4 1) 3, 
we see that — 3 increased by 1, that is, —3-+4+1 or — 2, being 
divided by — 2, the exponent of x in the binomial, gives asa 
quotient a positive whole number; we suppose, therefore, 


z—l : 


—2 2 ee —2 
a? a? +1 =z, whence x7? = - 5 


a? 
and as a *d« is, excepting a constant multiplier, the differential 
of a—*, we differentiate, which gives 


whence we deduce 2~°da=— The differential 


at) 
ata d x (a? ere ET), 
is therefore changed into 
ms 
—adz _-3 —z *dz 
2 a2 bee 
of which the integral is 


2(1—3) 
or, substituting for z its value 


84 | Integral Calculus. 


(a? + op? \2 
1 x 
aca Oat Ts 


Thus the process for integrating is the same in this case as in 
the preceding. 

93. We have supposed hitherto that the power of «# was found 
in only one of the terms of the binomial. If it should occur 
in each, we should reduce the quantity to a form in which 
it should appear in only one of the terms, by dividing the bino- 
mial by one of the two powers of a occuring in its terms, and 
multiplying the quantity without the binomial by the same power 
raised to the power indicated by the exponent of the binomial ; 
and that, for the reason just given (93), in rendering the exponent 
negative. ‘Thus, if it were proposed to integrate 
a nae as ert da (ae 4 uty 2, 

ai 1 
we should change it into a? a—1(a#) ?da(a+-x) *, by dividing 
the binomial by 2, and multiplying the quantity without by @ rais- 
edto the power — 3, which is that of the binomial. This quan- 


tity is reduced to a? ada (a + a), If we should apply to 
it the rule of the first case (91), we should not find it integrable, 
but by rendering negative the exponent of & in the binomial, we 
have 

oa teat —1, \-3 2 i ee 
atx? (7) *dx(ac +1) 7, or atx “da(ax +1) 2, 


which (93) is integrable. Making, therefore, a a—?+41= 2, we 


Z— 1 | OCs 
have (as -; differentiating, we have 
dz —adz 
—a*dx= —, or o2dx= ; 
a a 
therefore 
F my . 1 Le 
a2 ada (axz-'+1) ?=—adz.z *=—az *dz, 


f . az! 1 
of which the integral is ——— +. C, or —2a2z? + C, or, restor- 


ing the value of z, 


Quadrature of Curves. 85 


ard a ed 
=—2 i 1)2 : 
Sez) a(aa~*41)?4C 


“a 
=—2af* 41 + €, 


If, upon the examination of a binomial differential, it is found 
to be comprehended in neither of the two cases above mentioned, 
it is useless to expect an integral purely algebraical. 

As to polynomial differentials, that is, those whose complex 
quantity contains three or more terms, they are integrable in the 
cases mentioned in art, 85, &c. ‘There are also some other cases 
in which they admit an algebraic integral, but they are very few 
and rarely occur. We shall not, therefore, occupy ourselves with 
them at present. 

We shall hereafter show the method of discovering those which 


are integrable, and those whose integral may be referred to a 
given integral. 


Application of the preceding rules to the quadrature of curves. 


94. In order to find the area or quadrature of curve lines, we 
consider these lines as polygons of an infinite number of sides ; 
and from the extremities Mand m of each side (fig. 34) we 
imagine the perpendiculars MP, mp, let fall upon the axis of the 
abscisses, dividing the surface into an infinite number of infinitely 
small trapeziums. ‘Then we consider each trapezium, as Pom, 
as the differential of the finite space APM; because in fact 
PpmM =Apm—APM=d (APM) (6). It is therefore only 
required to express algebraically the little trapezium Ppm/M, and 
then integrate this expression by means of the preceding rules. 

But, in considering P pm M as the differential of the surface, it 
must be observed, that it is no more the differential of the surface 
reckoned from 4, the origin of the abscisses, than it is the differ- 
ential of any other surface KPIZ reckoned from a fixed and 
determinate point K; since we have equally 


PpmM=Kpm L— KPML=d (KPML). 
When we integrate, therefore, we have no right to refer the integral 
given directly by the calculation to the space 4M, rather than to 
any other space KPL.M, which differs from it by a determinate 
and constant space KL. We must therefore add to the integral 


Fig. 34. 


86 Integral Calculus. 


found by the calculation, a constant quantity expressing what the 
space proposed to be determined differs from that given directly 
by the calculus. It will be seen in the following examples, how 
this constant is determined. Let us, in the first place, find the 
expression for the space P pm JU. 

Calling 4P, 7; PM, y; wehave Pp=da, pm=yt dy. 
The surface of the trapezium P pm M is (Geo. 178) 

so AS Be inate A ee 
2 2 

But to indicate that dy and dw are infinitely small, we must re- 


xdx=ydxe+——_— 


ject ave *, which is infinitely small, compared with y dx; we 


have therefore yd a as the general expression of the differential of 
the element of the surface of any curve. 

In order to apply this formula to a proposed surface, whose 
equation is given, we must deduce from this equation the value of 
y, which we substitute in the formula y da, we then have a quan- 
tity in terms of w and da, which, when it can be integrated by 
the preceding rules, will give, with the addition of a constant 
quantity, the expression of the surface of this curve, reckoned 
from any point we please. We have then only to determine the 
constant, which is done by expressing from what point we choose 
to estimate the surface. We shall now illustrate this theory by 
some examples. 

Let us take, for the first application, the common parabola, 


; ae Oe eS 
which has for its equation y? =pa. Wehavey=4/pa=p® a?; 


aaeu 

therefore ydx ale 2? da. 
1 

pri wt dav 


3d x 
this last expression then 1s that of the surface of the parabola ; so 
that, knowing the absciss 2 and the parameter p, we shall have 
the value of the space 4P.W ; or, of the space KPL.M estimated 
from a determinate point A, if the constant C be determined, that 
is, if this integral express actually from what point we estimate. 

Suppose, then, that we wish to estimate the space from the 
point 4; then we have 


APM = 2p? a3 + C. 


But (83) fp? aha + C=2p?e 40; 


Quadrature of Curves. 87 


In order to know what becomes of the constant on this hypothe- 
sis, we observe, that when x becomes 0, the space 4PM is also 
zero; in that case, the equation is reduced to 0 = 0 + C; there- 
fore C = 0; in order then that the integral may express the space 
estimated from the point 2, the constant C must be zero; that 
is, we have no constant to add, and we have, generally, the in- 


definite space 4PM = 2p? x2, 
But if we wished to estimate the space from the point K, such 
that 2K = 6, (b being a known quantity); we should have 
KPLM =2 p? «? +€; 
now this space KPL.M becomes zero, when AP or x =b;3 we 


: aes 
have therefore, in that case,0 = 2 p? 624; therefore 


and consequently 
ate Ss 1 73 
KPLM = 2p? x% — 2 p® 0%. 
We thus see what purpose is served by the constant, which is 


added in integrating, and how the conditions of the equation alone 


can determine it. 
3 


OE =Y5 
1 


therefore 2p? w?, or 3 2p? x? xX «=2y%3 since therefore 


i i 
2 2 


4 i 
We observe that 3 p x essbut pir 


2 p a2 expresses the Sree APM, this space will also have for 
its expression 2 x y, that is, 2 4P x PM, or 2 of the rectangle 
APMO, whatever AP may Be 

In like manner 

ae Liga 
OE ati pe dr Ae 

but, when x => AK=4, the equation y2 = p 2 gives y? = pb, 
and cone i a = p? b? that is, KL = p? b ; therefore 
2p? b? or 2 2 3 2 x b=2 KL x AK; therefore since the 
space KPLM is Rccaaa by 2 py? oF 2p? B3, it will 
have also for its expression 3 4P x PM —2 AK x. KL, that is, 
2 APMO— 2 AKL. 

The parabola is the only one of the four conic sections suscepti- 
ble of being squared. 


Fig. 35. 


88 Integral Calculus. 


Let us take, as a second example, parabolas ofall kinds, whose 
general equation (30) is y™ +" = a™ a* ; 


we have 
™m n 
y ee OTe Ly th qatn genet ts 
then 
m n 
yd x = qgmtn ymtrn da : 
and 
7d 113 44 m nN + 
gamtn gmpne! m+n mtn men? 
0 as Ht x C 
Sy n Nn © m+2n : 
m+n 
™m n 
m+n aia i) m+n 
— a men m+n ws, Ae Ge 
m+ 2n # + m + 2n vo: 


So that if we wish to estimate the space 4PM from A, the origin 
of the abscisses x (fig. 35), which requires the integral to be zero 
when 4PM is zero, and when consequently # is zero, then the 
constant C is zero, and we have simply 
m+n 
APM = ECE IOT TS rt al! 

that is, the space 4PM is always a determinate portion of the 
product xy or of the rectangle 4PMO, it is that portion of it 

n+ 7 
m+ 20 
the values of m and n: that is, on the degree of the parabola. 
Thus all parabolas are susceptible of being squared. 


expressed by the fraction , the value of which depends on 


It will also be found that all the hyperbolas referred to their 
asymptotes, (except the common hyperbole,) are susceptible of being 
squared. But as, in the determination of the constant, we sometimes 
find an indefinite quantity, it may not be useless to examine here, in 
what sense it is to be understood. Let, therefore, y™ = amt" g—" 
be the equation of these curves ; we have 

(ake 9 
ice Wi Be oo 


therefore 
Men —t 
AE Sot hee 

and 

m+n fares 

Oa m ad m+n gin 

fyda= ——- ar ve % a C;: 
yates ade o_o 


Quadrature of Curves. 89 


a quantity in which there is no difficulty in determining the constant 
when mexceeds n. But when m is less than n, we find an infinite quan- 
tity for the constant, if we estimate the space from the origin of a; and 
a finite quantity, if we estimate from any other point. Let us sup- 
pose, for example, m = 1, and n = 23 in which case the equation 18 


y =a} a—*; the surface is then reduced to—a x—-*+ C,or C— —. 
If then we wish to estimate the space from 2, the origin of the ab- 


a as ri as 
scisses z, C— — must be zero, when « = 0; that is, C— ithe 0, and 
x 


3 . . e . 
consequently C == ; that is, is infinite. If, on the contrary, we 


wish to estimate the space from the point A, such that Kil = 6, we 
3 


3 
have C — “y == 0, which gives C = —, and consequently this space 


b 
Nama ha ; ; : : 
contd NEBR This result is to be understood in the following man- 
ner. 


. : ; as 
The curve which has for its equation y = a5 a—? or y = —> ex- 
wv 


tends to infinity along the asymptotes 42, AV (fig. 36); but ap- 
proaches much more nearly to the asymptote 4Z than to the asymp- 
tote 4Y; for, when 2 is infinite, y is infinitely small of the second 
order, but when y is infinite, x is only infinitely small of the order 1; 
if therefore we estimate the spaces from the asymptote AY, they are 
infinite, because the space comprehended between this asymptote and 
the infinite branch BS is infinite. On the contrary, the spaces com- 
prehended between the branch BM and the asymptote 4Z to infini- 
ty, have a finite value, because after a very short interval, the branch 
approaches its asymptote very rapidly, so that the infinitely long 
3 3 
space KLUOZ has for its expresssion oa and PMOZ = <j and con- 


3 3 
sequently KLMP = ; rel ~ Whence it follows, that though we 


may not obtain the spaces estimated from .4V, we may, nevertheless, 
find the spaces KLMP, estimated from a point &, taken as near as 
we please to AF. 


Let us take, as a third example, the curve which has for its 
a% o— x 


equation y= . za and which will be found to have the fig- 


Fig. 36. 


ure delineated (fig. 37), by giving successively to x arbitrary val- pig 37. 


ues, and a determinate value to a. 
We shall have then 
at*xdx—aidx 
a2 


© 


ydu= 


> 


12 


Fig. 38. 


90 Integral Celeulus. 
and (83), 
fyde=APM = a 40; 
a 
and if we wish to estimate the space 4PM from the point A, the 
origin of the abscisses a, this integral must become O with vw, which 
shows that the constant C= zero. So that the indefinite space 

gy Real a 2a? x2? — x» 
APM is simply ach 
y is composed, as ia the present case, of simple terms only, it 
will always be easy to find the surface. 

95. We have seen} how, by the assistance of algebra, we may 
imitate any perimeter whatever BCD ( fig. 38), by causing to 
pass throvgh a ceriain number of its poinis 2, 6, C, D, a curve 
line whose equation shall have the form 

yra+tba en? Leln a fa*, &c., 
and how, to this ead, we may determine a, 6, c, &c. Let us now 
suppose that it were required to find the surface ABCDLK, al- 
though we had not the equation of the curve 260. 

We should draw through a certain aumber of points 4, B, C, D, 
a curve 2 e B/C g D, which would coincice with the former the 
more perfectly according as we should take a greater number of 
points. And, as we should then have the equation of this latter 
curve, we might consider ti as the equation of the curve ABCD, 
at least for the exteat ABCD. Bui this equation being then of 
the form y= a--b~-+-¢ x? +ex, &e. 
of which all the terms are simple, we should easily find the sur- 
face according to what has just been said. We may apply this 
method to the measure of ibe surfaces of the section of vessels. 


. Aad, in general, if the value of 


In general, it may be applied to find, by approximation, the surfaces 
of curves and the approximate integral of quantities which cannot 
be integrated exactly. Indeed, every differential may be regarded as 
expressing the element of the surface of a curve, whose ordinate is 
equal to the total factor of dx. For example, dv ./a2+-z2_ is the 
element of the surtace of the curve, which has for its ordinate 
ya= /a2z+22- Thus, calculating by means of this equation some 
values of y for certain values of «, and drawing through the extremi- 
ties of these ordinates a curve of the nature of those just mentioned, 
the surface of these latter being found, would be the approximate 
value of the integral of d x a2 22 for the extent of x. 


~ 


t See note at the end. 


Rectification of Curves. 91 


Let us take another example. It shall be that of the surface 
of the curve which has for its equation a° y? =x* (a3 —#$), 
this equation gives 


i (a3 — 28) _ x? ie 
®! a> a? fa 
therefore (taking only one of the values of y), 
2 2d ux 1 
yd x= Se Vee = ( — as) ; 
now this quantity is integrable (89) because x? d «x is the differ- 
ential of the term 2° in the binomial, divided by the constant 
number 3. 
Therefore (89) we have 


V/s — 23; 


y= oe 


3 3 
_ «dx (a3 —x3)? Hitie Oat — x3)? C 
fyde= 3a% fa .—8arda vane 9a? fa eos: 


With regard to the constant C, it will be determined by decid- 
ing from what point we wish togeckon the surface. 


We may also find the surface of curves by decomposing them into 
triangles. instead of trapeziums. For example, we might find the 
surface of the segment 4VQ ( fig. 34), by considering it as composed Fig. 34. 
of an infinite number of infinitely small triangles, such as 4Qq, 


This triangle would have for its expression sik a by letting fall 


the perpendicular Q¢, or, which amounts to the same thing, by de- 
scribing from the centre 2, and with the radius 4Q the infinitely 
small arc Qt. Then calling 4Q,¢, and the arc Qt,d, we should 
have 2g =t + dt, and conseyuently the triangle 


AQq = fe dx =o" 4 SESE that is ie ae 
dtdx 


rejecting the term ,1n order to indicate that d« and d¢ are in- 


finitely small. Nothing more would be required than to have the 
equation in terms of x and ¢, in order to substitute in place of ¢ its 
value in x, and integrate. 


Application to the rectification of Curve Lines. 


96. To rectify a curve line is to determine its length or to as- 
sign a straight line which shall be equal to it, or to any proposed 
arc of it. The following is the method to be used, when the rec- 
tification is possible. | 
Considering the curve AM (fig. 34) as a polygon of an infi- Fig. 34. 


92 Integral Calculus. 


nite number of sides, the small side Mm may be regarded as the 
differential of the arc 2M, because 
Mm = Am—AM= d (AM). 
Now, drawing Wr parallel to .42P, we have 
M m = /Mr2 rm? = Vdz2 + dy? 
it is therefore only necessary to integrate 4/722-.dy2. In order 
to this, we differentiate the equation of the curve, and having de- 
duced from it the value of dy, expressed in terins of w andda, 
or that of da in terms of yand dy, we substitute it in the ex- 
pression 4/dz2 —dy2 , which will thus contain only terms of w and 
dx or of y and dy; we then remove dz? or d y* from under the © 
radical sign (.4/g. 130) and integrate. 
To give an example, let us take, among the parabolas expressed 
generally by y™t" = a™ a", that which has for its particular equa- 
3 


y* y? 
tion y* = a «?, we deduce, x? ==—; and«=*, ; whence 

”- oe 

yidy eo ydy? 

dx=2 7 J andda?=—22°! , 

as 4 a 
therefore 
Sie Fax = fay? Sas le = dy ein 


Now this quantity is easily integrated (90), since the exponent of y 
out of the binomial is less by unity than its exponent in the bino- 
mial. ‘Therefore 


ee 


The constant € is determined as follows. If we wish to estimate 
ihe arcs AJM from the point 4, the origin of y, the integral or 
value of the arc .4.M@ must become zero at the same time that 
y=0. But when y= 0, the integral is reduced to 

Sa 3 _ 8a 


therefore tie + C=0; whence C = — >" Therefore thé 


Curve Surfaces. 93 


length of any are 4M, reckoned from the vertex J, is 


3 
8a 1 9y\2 8a 
27 4a Q7 


If it be required to find what other parabolas are susceptible of be- 
ing rectified, we may proceed in the following manner. The equa- 
tion y+" = a” a” , which belongs to these curves, gives 


™m n 
y —_ gern x m+n, 


Let us take us 


n 

= k, and ——— =»; we have y = a* af: 
m+n ‘ m+n ?? y J 
whence 


dy =pak xe—1dx, and dy? =p? a2* x ??-2d x2; 
wherefore 
JT FTP = TPS pe dae 
=dx L + p? a2k 2p —2, 

a quantity which is not integrable in this state except when 2p —2=1. 
But, if we change the sign of the exponent of » under the radical 
we have ROS NE A/T Bi? a? ®, 

which (91) is integrable if p — 1 increased by unity and divided by 


— 2p + 2, gives a positive whole number 3 that is, if apreray = t, 
t being a positive whole number. Whence we deduce 
2t 
but p = ——, wherefore — Lt Soa. whence m = ——; 
eed? ee an Peet fl Ce ate 
thus the parabolas which may be rectified, are those comprised in the 
Q2nt+n n i 
equation yo Ft = art we 3 
or, extracting the root of the degree n, 
2t-+1 1 
"09 3) Ra aad a 


t being any whole number whatever. 


Application to Curve Surfaces. 


97. We shall confine ourselves to the surfaces of the solids of 
revolution. We call by this name the solids conceived to be 
generated by the revolution of a curve 4M (fig. 39) about a Fig. 39, 
straight line 4P. 


94 Integral Calculus. 


We imagine that while 4.1 revolves about AP, the small side 
Mm describes a zone or portion of a truncated zone, which is 
the element of the surface, and which (Geo. 581), is equal to the 
product of Mm, by the circumference which has for its radius the 
perpendicular let fall from the middle of Vim upon 4P, or which 
is the same thing, since .1/ m is infinitely small, by the circumfer- 
ence which has for its radius PM. Now the are 

Min = ya? ay? 3 
and if we represent by 7: c the ratio of the radius of a circle to 
its circumference, we have 7+ : ¢ :: y is io the -circumference 


ae Se le ‘ ct 
waich has PM for its radius, which will be fy. whence we 
7 


Cc pcb ese ae as 
have —# ait + by? e 


for the element of the surface of the solids of revolution. 
98. Lei us suppose, as aa example, that the surface of the 


- sphere is required. ‘The generating circle AMB (fig. 40) has 


for its equation y? =a x2 — x, calling 4P, x, and PM, y. Whence 


2 ost ws tZadx—adzuz 
Y= Yor 43, and oy ee 


A/ 4X — x2 
whevrefoie 
dy? _tabdx?—axdar+xz2 dx? 
Hea ax— x2 : 
therefore 
t. Ty hat do? aed 22 ot ad ot 
Ro at dcx acd«? +2%*dzx 
ad hed d Qype — d x pelt SR 0 a a 
Ris + Y + i Ree 
= foe dat -~ e2dx?+4+1 qa? d.x?—a xd 22+ 0? d Pe 3 ada 
aOXx— x2 ~ fax —x2 


Sider) . } 2. ONS a AEE 
Substituting, then, in the formula td 22 —dy2, for y and 


Vda + dy?, their values just found, we have 


“ar — x dada 
er ee So 
n° / ax — x? 
1 
: acdax 
which becomes a ae and 


zacdx sgace C 
(ibys! 1. oT ae an 


d gacx , : 
or simply Tae if we reckon the surface from the point 4. 


ZaCx Zac 
2 2 

> or 
r yr 


Now x, expresses the surface of a cylinder having 


Curve Surfaces. 95 


for its base a great circle of a sphere, and for its altitude, a ; 
Wuich agrees witn what has already been found (Geo. 523). 

99. ‘Io find the surface of the paraboloid, (which is the solid 
geaerated by the revolution of the parabola 2 JM (fig. 39) about Fig. 39. 
its axis), we have the equatioa y? = p «3; whence 


2 2yud 4 a2 2 
eee ond eas ee ee 
P P PP 


whence 
Os rh, ca hae Te aaa 
a/ dx? + TP= lay ee —d y in I: ro ; 
wherefore 
(om! gies Marte yew cy di aay? 
oda pap = it a lg 
opp 
which, being integrated, (90) gives 
cy dy coat Gh ay La) 
+0 =45*( , sy } 
Soy dy TF YD — tibiae iC. 


PP 
Now, in order that this quantity should express the surface reck- 
oned from the vertex .7, tt must become zero, when y = 03 wut, 


; Pee NS ager DC : 
in that case, it becomes 7=—. (1)? -1. C, or ——— +. C 3 waeace, 


12% 
pict Pee: 
ee C= 0; that is C=— alk: 
whererore, the surface of the yess paraboloid 2.MUI.A4 is 
BACK eo see 
lar tp p? Tr 


Applicotion to the measure of solidity. 


100. tn order io measure the solicity of bodies, we may 
imagiae them composec of infinitely thia parallel segments, or of - 
an infinite number of pyramids, whose summaiis unite in the 
same point. In the first way of viewing the subject, the difference 
of the two opposite surfaces, which terminate each segment is 
infinitely small, and must consequently be omitted in the calculus, 
if we would indicate that the segment is infinitely thin. Thence 
follows that we must take, as the expression of the solidity of this 
segment, the product of one of its opposite bases by its infinitely 


Fig. 41. 


Fig. 39. 


Fig, 41. 


Fig. 39. 


96 Integral Calculus. 


small altitude. If, for example, we consider the pyramid SABC, 
(fig. 41) as composed of infinitely thin segments, like abcde f; 
we may take, as the measure of this segment, the product of the 
surface abc ord ef, by the thickness of this segment. 

In like manner, if we consider the solid generated by the revo- 


lution of the curve 4M about the straight line 4P (fig. 39), as 


composed of infinitely thin parallel segments, we must take as the 
measure of each segment, the product of the surface of the circle, 
which has for its radius PM, by the thickness P p. 

This principle being laid down, we thus estimate the solidity of 
the whole body. We consider each segment as being the diffe- 
rential of the solid, because the segment JM m 1 L is in fact 

=AmlA—AMLA=d (AMLA) ; 
and having determined the algebraical expression of this segment, 
we integrate. 

Let it be required, for example, to find the solidity of the pyra- 
mid SABC ( fig. 41). Supposing the surface of the base ABC to 
be equal to the known quantity 6°, and its altitude St=h, we 
represent by x the distance St of any section; which gives dx 
for the thickness of this segment. ‘The surface abc is found by 
the following proportion (Geo. 409) ; 

STE See Be. ables 


that is, 
pede 
keg oR ES CE pad ote 2 ees eat 
thus the solidity of the segment will be 
ake is Kees d De ae Git ie. D> L208 C pale oi 
ban eae Rage. Wi08 h3i sh ca Ba? 


if we reckon the solidity from the vertex S. This quantity, which 
expresses the solidity of any pyramidal portion S abc, is the same 

hb? x* x ; i St , % } 
as X Fe which is a be x mite which agrees with what has 
been demonstrated (Geo. 416). 

101. As to the solids of revolution, we may find a general ex- 
pression for the elementary segment or differential. For, suppos- 
ing that 7 : ¢ expresses the ratio of the radius to the circumfe- 
rence, we shall find the circumference, which has PM ( fig. 39) 
or y for its radius, by the following proportion, 

AEB 


Wy) wie <8. 8 
r? 


Curve Surfaces. 97 


c . he . 
if we multiply this value — of the circumference, which has PM 


: cu 
for its radius, by3 y or half of the radius, we have oo for the sur- 


face, which, being multiplied by the thickness Pp or da, gives 
éy*dax 
2r 
solid of revolution. ‘To employ it in any particular csse, we have 
only to substitute instead of y its value in terms of x derived from 
the equation of the generating curve 4M, and integrate. 
102. Take, as an example, the spheroid generated by the rev- 
olution of the ellipse fa its major axis (fig. 42). ‘The equation Fig. 42. 


for the expression of the element of the solidity of every 


of the ellipse is y2 = -> (ax — 2) calling 4P, x, and PM, y, 
and the axes AB and Da a and 6; we have then, 


ap Y 2 2 
cy OF = £" (ax—a") daa s, (axdx—x2*da), 
of which the integral 
_ cb* fax? 3 sublet ax x3 
Seale s)t =25 (> SPREE | 


when the solidity is reckoned from the point .4. 
In order to obtain the entire spheroid, we suppose x = 4B=a, 


c b? as as cabh* 
and.we havej-—--; pays oS G Tats 7) which is reduced to - ORT. 
hich is th thing as gO oes hab th a 

whicn 158 e same Ing <— x $a, or— x 2a; now 


expresses the surface of the circle, which has b or Dd for its di- 
c b? 

ameter; and ap xe would consequently express the solidity of 

the cylinder circumscribed about the ellipsoid. Therefore, since 


i. ) ae b? 
the solidity of the ellipsoid is here — x $a, we conclude that 


it is two thirds of that of the circumscribed cylinder. And as the 
sphere is only an ellipsoid, of which the two axes are equal, the 
sphere is therefore 3 of the circumscribed cylinder, which agrees 
with what has been demonstrated (Geom. 549). 

103. If we wish to determine the solidity, reckoning from a 
determinate point K such that 4K =e ; we take the general inte- 

2 8 

gral ae (“S-5) + ('; and as the solidity is to be reckoned 


13 


Fig. 39. 


98 Integral Calculus. 


from K, this integral must become 0 at that point, that is, when 


. . cb? /fae* e3 
e2=e; in this case it Ae ‘ 
| ; become a(S 5) +03 


eh? fae* e3 
2ra? (F —=)+ Re 


and consequently 


‘hee cb? a e2 es 
~~ 8raz7\Q 3 J? 


thus the solidity, reckoned from the point K, has for its expres- 


sion 
c hb? ax oh c b2 aez  e@8 
Q2rar \ 2 3 2raz\ 2 B38 S° 


Such, therefore, is the value of the segment of an elliptical sphe- 
roid, comprehended between two parallel planes, perpendicular to 
the axis, and at the distance x — e from each other. 

We may, by this formula, calculate the solidity and consequently 
the weight of the masts and yards of vessels, which are portions of 
elliptical spheroids. ‘I'he same formula serves also to measure 
the capacity of casks, whose external surface may be regarded as 
a portion of such a spheroid. 

104. Let us take, as a second example, the paraboloid (fig. 39). 


whence 


2 
The equation of the parabola is y? = pa, thus the formila~ tS ee 
d : : 
becomes Fi Saal is whose integral is 
pads © 
bi wine aks SS Ape ae cy? 
4r Ce eget oy in a ees xs+6, 


by substituting for pw its value y?. If we reckon the solid from 


the point A, as it is zero when «= 0, the constant C must be 


zero, and the solidity is reduced to ae x =3 now lt expresses 
the surface of the circle, which has P.M for its radius, that is, the 
base of the paraboloid 4MLA; therefore the paraboloid is half 
the product of its base by its altitude ; it is therefore half of the 
cylinder of the same base and altitude. 

If we wish to reckon the solidity from the known point K, such 
that 4K =e; then, as the solidity must be zero at the point K, 


that is, ae i me ki the general Bo must be zero at the same 


time; thati ise i sig C, Beromine = er C, equals zero, 


i. 


Measure of Solidity. 99 


wherefore 
e2 
ie +C=0, 
and consequently 
. roca cp e* 


4r°? 

whence, the solidity of the segment of a paraboloid comprehend- 
ed between two parallel planes, whose distances from the vertex 
Greener ¢ 
4r 4r 
the excavations of mines. 

105. We may take, as another example, the hyperboloid or 
solid generated by the revolution of the hyperbola about one of 
its axes. We may also take the ellipsoid generated by the revo- 
lution of the ellipse about its lesser or minor axis, which is called 
the Flattened Ellipsoid (Oblate Spheroid). That generated by its 
revolution about the greater or major axis is called Elongated 
Ellipsoid {Prolate Spheroid). We should, in like manner, find 
that the flattened ellipsoid is 2 of the circumscribed cylinder : 
that is, that a and 6, being the greater and less axis of the gene- 


. This formula may serve to estimate 


are x and e, is 


é : slide b? 
rating ellipse, the elongated spheroid has for its solidity ——, and 


. 2 
the flattened spheroid has for its solidity soos thus the elongat- 


cab? carb 


ed spheroid is to the flattened spheroid: : AG arte 


Ws 


as the lesser axis to the greater. 
We shall now leave the solids of revolution. 
But in order to accustom beginners to extend the application 
of these methods, we shall give two additional examples. 
106. In the first, it is proposed to find the solidity of a cylindri- 
cal angula formed by cutting a cylinder by a plane oblique to its 
base, and which, for greater simplicity, we will suppose to pass 
through the centre. It is the solid JDBE represented (fig. 43)« big Ae 
If we conceive this solid to be cut by parallel planes infinitely 
near each other, and perpendicular to the base AEB ( fig. 44), y., 44 
the sections will be similar triangles, whose surfaces will conse- ~ 
quently be as the squares of their homologous sides. Thus, cal- 
ling CE the radius of the base, r, the altitude DE, h, and PM, 
the base of the triangle PMN, y, we have 


Fig. 45. 


106- Integral Calculus. 


CAD BMNae ines: y* 5 


now 
rh 
therefore 
rhy* hy? 
PMN = 2rr 2r?’ 


ealling, therefore, 4P, x, which gives dx for P p, the thickness 
of the segment comprehended between two contiguous planes, we 
hy*dx 


) 
~r 


have for this segment. Now y is an ordinate of the circle 


which serves as base, and we have consequently y2 =27r «2 —2a?. 

hdx(2ra—x’? 

The elementary segment becomes then ie Aenee) 
h ; 

o (2rad«—.x?d x), of which the integral, reckoning from the 


, or 


Ai ad 3 ( 
point 4, is — (7 x? — ~). Therefore, to obtain the whole 
solidity, we have only to suppose x = 27, which gives 
h 8rs hr 
Peal x Cae eee 3 pM IMR AM SOY 0 et 4 
: (47 s-) =$hr? = x $r=CED x $AC 


2r 
= CED x 3 AB, 
ihat is, two thirds of the prism, which should have the triangle 
CED for its base, and the diameter 48 for its altitude. This 
may serve to measure the solidity of fortifications. 

107. As a second example, we shall investigate the solidity of 
a segment of an elongated ellipsoid, comprehended between two 
planes parallel to each other and_to the greater axis. 

Before proceeding in this investigation, it must be demonstrated 
that the sections of an ellipsoid, made parallel to the greater axis, 
are ellipses similar to the generating ellipse of the solid, that is, 
that their axes have the same ratio to each other as the axes of 
that ellipse. 

To this end, we conceive the ellipsoid cut by a plane, which, 
for the sake of preciseness, we suppose vertical, and passing 
through the greater axis 4B (fig. 45). The section will be the 
ellipse 4.DBE equal to the generating ellipse. We also conceive 
the ellipse cut by three other planes, of which two are vertical 
and the third horizontal. Let the less axis of the ellipse DE and 
its parallel MM’ be the intersections of the two first with the plane 


Measure of Solidity. 101 


ADBE, and ST that of the third with the same plane ADBE. 
This done, we say that the section of the ellipsoid by the plane 
represented by ST’ is an ellipse similar to DBE. 

We conceive perpendiculars to the plane ADBE to be raised 
from the points O and &, meeting the surface of the ellipsoid. 
These perpendiculars will be at once ordinates of the section 
made by S7', and of the circular sections made by M.V and DE, 
Now since they are ordinates of the circular sections, if we call 
the perpendicular at the point A, z, and the perpendicular at the 
point O, t, we shall have 

2=> DR. KE, 

t? = MO. ON. 
But, calling CD,1b; PM,y; CA,i4a; and CR= OP, u; we 
have 

DR=146+4+u,RE=1b—u, MO=y+u, ON=y—y, 
so that 

DR. RE =1 6? —u? =z, 
MO. ON= y? —u? =1?. 
But, by the nature of the ellipse (4p. 123), we have 


b2 
y= AR Na Za? — 27), 
calling CP, 7 Andk a Eppentine the ordinate SR to the less 


axis, we have (4p. 122) k?= 5 (4 6? — u?), whence we de- 


2 


2 k ° . 
>; substituting these values of u* and y? 
a 


duce u? = 1 6? — 
in those of z? and t?, we have 


b? 12 ost ot Vi 
z* = — Taait and, (aie : - 


a> az. 

whence it is evident that 

2? erp pats 2 sl lai sa aio k2 > k? —a? :: SR? 
or SR.RT:SO.OT; 


that is, the square of the ordinate z, corresponding to the point 

R, is to the square of the ordinate ¢, corresponding to the point 

O, as the product of the two abscisses to the first, is to the pro- 

duct of the abscisses to the second; the section made by ST’ is 

therefore an ellipse. 

b2 ke? 
2 


: bik >. 
Besides, the equation z? = , Or 2 = ——, gives 


mes kets b is as 


102 Integral Calculus. 


now 2, or the ordinate to the point R, is the semi-minor axis of 
this ellipse, and kor SR is the semi-major axis ; the two axes of 
this ellipse have therefore the same ratio as those of the generat- 
ing ellipse; and as nothing in the course of this reasoning deter- 
mines at what distance CA this section is supposed to be made, 
the same thing takes place for every section parallel to 2B. 

This being determined, if we wish to have the solidity of any 
segment of an ellipsoid, comprehended between the two parallel 
planes represented by 4B and ST’, we represent by S the sur- 
face of the generating ellipse; and since the ellipse, of which 
ST is the greater axis, is similar to this, we shall have the surface 
of this last by the proportion 


S k* 
A LIGE. s cases Tar? 
multiplying this surface by the eal ae thickness ft r or 


edu for the value of 


du of the elementary segment, we have ae 


ob 
this segment; but, according to what has just been said above, 


2 7 (132 2 
we have ha a Ge —wu?); 
whence the elementary segment will be 
Sdu (1b? —u? S 
| Mp Gaede: 3 = Fr Gb du—w du), 


24—4u*),if we reckon from 


+ 
the centre C. But if we reckon from the point K, the integral 
s : 
will be To 16% u—iu8)+C. 
In order to determine C, we call CK, e; then the integral must 
be zero at the point R where u=e; we shall have, therefore, 


— (a Ph € wee) ab = 9, 
4 
and consequently 
C=— (1 6? e— ze), 


+ s 
wherefore every segment of an elongated ellipsoid, comprehend- 
ed between two planes parallel to the greater axis, has for its ex- 


pression 


s s 
Ct tsk ee ama 


(4 b?e— Je), 


Measure of Solidity. 103 


er (1 68 u— 6? e— yu +363). 


Now 162? 4y—1b%e=15? (u—e). 
In like manner, 


1 whee 
1¢3 —1y3 —— (e? +eu+u?). 


Moreover, u—e represents the distance of the two parallel planes or 
the altitude of the segment comprehended between them ; if, there- 
fore, we make u —e=h, calling this altitude 4, and substitute for 
eits value w—A drawn from this equation, we have, after all reduc- 


3 
tions, oe G b2h—hu? +h? u— =) 
yr re) 
Sh Sh®* h 
or oa Guat os 
2 
but we have had, above, k? = — 46% —u?*), and consequently 
b? kc* 
4 6? pny U2 — — rc » 


The value of the solid segment is therefore changed into 


Shk? | She hk 
iaa + ibd ( 3)" 

kk 

ss for the expression of the surface of the see- 


But we found . 


_ 
tion made by ST’, or the inferior section; calling then this sur- 
face s, we shall have 
Sh? h 
sh + Ibo u—z)s 
finally, if we represent by s’ the surface of the section made by 
LK, and by / its semi-minor axis, we shall have, from the simi- 


. . h2 ¢! 
larity of the sections, 36? : 1? :: S:s’; whence S= 1 at 


e 


which gives, for a final expression, 


s’ h” h 
That is, we must, Ist, Multiply the surface of the less section by 
the altitude of the segment ; 2d, multiply the surface of the great- 
er section by the ratio a of the square of the wtitude of the seg- 


ment to the square of the semr-minor axis of the superior section; 


104 Integral Calculus. 


and by the distance from the centre to the infertor section minus 
a third of the altitude of the segment. 

This rule may be usefully applied to measuring the solidity of 
that part of the hull of a ship which the lading causes to sink be- 
low the surface, whenever the figure of this part may be compar- 
ed to a portion of an ellipsoid. In this case, s will represent the 
section of the vessel without its lading, at the surface of the 
water ; s’ the section with the lading; A the distance of the two 
sections; 2 the greatest breadth of s’, and wu the distance from 
the greatest horizontal section of the spheroid, to s. 

As to the mode of measuring s and s’, it may be observed, that 
one of these surfaces is determined by the other, since, belonging 
to similar ellipses, they must be to each other as the squares of 
their greater or less axes. It is therefore only necessary to know 
how to determine one of them. Now, we shall presently see, that 
the surface of an ellipse is to the surface of a circle which has 
for its diameter the greater axis of the ellipse, as the less axis is 
to the greater. As we know then how to estimate the surface of 
a circle, at least to as great exactness as may be desirable, we 
shall easily determine that of an ellipse whose axes are known. 


On the integration of quantitres containing Sines and Cosines. 


108. We found (21, 22) that 

d (sin z) = dz cos z, 
d (cos z) = —dzsin z; 

therefore, reciprocally, 

fazcos z=snz+C, 

—dzsinz =cos z+ C. 
It is requir ed to find the integral of dz cos 3z; we have 
3dzcos3 z sin Sz 


fazcssz—f 2 = LO} 


In like manner 


Sazsin8z— [AZZ s® —aeeenes oe 4 


re 


In general, m being any constant quantity, 


‘ —mdzsinm z — cos m z 
faz sin mz = i hm 


Let it be proposed to integrate (sin z)"d z cos x; because 
(sin z)"d z cos z = (sin z)” d(sinz), 


Sines and Cosines. 108 


we have 

(sinz)? +! 
n+ 1 

If the proposed differential were (sin mz)" dz cos mz, we should 

give it the form 


f (sin z\"dz cos z= + C. 


(sinm zx)" mdz cosmz (sin m z)"d (sin m z) 
a) Go re oar SRR ay oe 5 RT RT RTT GT PETES | 
m mm 


of which the integral is 

* (sin m zjrrt LC 
m(n + 1) ; 
In like manner 
(cosm z)®X — mdz sin mz 


—— Tt 


f (cos m z)"dzsinm z% = 


(corm zrtt 
— —m(n +1) sb 2. 
Let it be proposed to integrate d x sin p x cos gz, p and q being con- 
stant quantities. By what has been demonstrated (Tr. 27), 
sinpzcosqz =}sin(pz+qz) + 4sin(px—qz) 
= } sin (pP+q)~ +4sin (p—q)%; 
we have therefore to integrate 
tdzsin(p+q)z%+4dzsin (p—q)z 
Fe CB) dzsin(p+q)z 44 (p—q)dzsin(p—q)z 
nae: Ey Wee Pap eas 
P+4 Be 
of which the integral is 
—joos(ptq)z — zcos(p—aq)\z 46 
P+ Hie 
We should integrate in the same inanner dzsinpz cosqz sinr<z, 
by converting these products into the sines or cosines of the sum or 
difference of the arcs pz.qz,1 z, &c. (Tr. 27). 
If we wished to integrate d z (sin x)’, we should change this dif- 
ferential into dz sin z (sin z)? ; now 
sinz* = sin z sin z =40c0s (7 —z)—4 cos (x + 2) 
=1.0s o— 1 cos 2% = F— 3 c0s2z; 


therefore 
dzsinz* =1sinzdz—1tdzsin z cos 22. 
We should then resolve sin z cos 2z in thesame way that sin pzcos qz 
was resolved above, and the integration would be easy. We see, there- 
fore, how we might integrate d x sin 2”, n being any positive whole 
number. We should proceed in a similar manner to integrate 
dz cos z". We may theretore, on the same principles, integrate 
quantities of the form ) 
dz sin pz™ cos qz"sin rz , &c. 

m, n,s being positive whole numbers. 

Finally, these principles, what has been already shown (Tr. 27 &c.), 


14 


Fig. 40. 


106 Integral Calculus. 


and what has been laid down above on the integration of quantities, 
give us the means of integrating differentials affected by sines and 
cosines, whenever they have an algebraical integral ; and when tan- 
gents occur, they may be reduced to the differentials of sines and co- 
sin z 


sines, by observing that t Se pen 

i y observing that tang x = ——— 

On the mode of integrating by approximation and some uses of 
that method. 


109, ‘This has nothing to do with simple differentials, since 
they, as we have already seen, are always easily integrated. It is 
only for complex differentials which elude the methods already 
given. 

The art of integrating by approximation, consists in converting 
the proposed quantity into a series of simple quantities whose 
value continually diminishes ; each term is then easily integrated 
and it is sufficient to take a certain number of them, in order to 
obtain an approximate value for the integral. 

The rule given (A/g. 141) for raising a quantity to any propos- 
ed power, and which is equally applicable to polynomials, is the 
method we employ in order to integrate by approximation. We 
shall now give some examples. 

110. Let it be proposed to find the length of the arc of a circle, 
AM (fig. 40) by means of its versed sine AP. 

Supposing the arc M m to be infinitely small, if we draw Mr 
parallel to. 4P, and also the radius CM; the similar triangles 
CPM, Mr m, give 

PM 3. CM Stirs’ Mm. 
Making AP, a, and the diameter 1B, 1, we have Mr=dz, 
CM =i, and PWt=Vz—a2. Therefore 


COSTER sie 
eee 
and consequently 414 = [JES This quantity cannot be 
/x— x2 


integrated by the rules given above, wherefore we change it into 


Adx . me “3 
DE pecan (dlg. 123), and then into fa 2da (l—x«) ” 
x /1—¢ 


{PM = Nuc — CP — fla 2) = f/i—x?, 


Approximation. 107 
—1 
(Alg. 133). We then reduce (1 — x) ? to a series (Alg. 144) ; 
and find, after making all the reductions, 


pally 3 
(l—a) ?=1+jfr+ }u? + 0° +&c. 
therefore 
fet >da(1—2) raft ae “3 dy (Iti ope x45, - x4 &c.) 
=f{Ge Meee 7 Peel Apes det 8) 


3 


; i 15 as 

PoE Tm wh &c. 
as +3 x +43; x +455 Pee + &c. 

a quantity to which there is no constant quantity to be added, be- 

cause when « = 0, it becomes zero, as should be the case, since 

then the are AJM, which it expresses, is zero. 


1 F : 
By reason of the common factor 27, we may give to the series 
expressing the arc 4J/V1 the form 


a? (l+iev+3 ot? bats aw? + Xe.) 

We now observe that the seaen sine xv is always less than the di- 
ameter 1, except when we speak of the semicircumference, 
wherefore x is always a fraction, and consequently the values of 
the terms of the series will diminish, in proportion as the versed 
sine of the arc in question diminishes. If, therefore, we wished 
to find, for example, the length of the are whose versed sine is the 
hundredth part of the diameter, we shonld have a= +1, =0,01, 


1 
and consequently a? =0,1; we should have therefore for the 
value of this arc 


0,100 plas 4 pl + ital et 01)? )5 


and as the next successive term rae this series would be at least a 
hundred times less than the last of those given, since each is more 
than a liundred times less than the preceding, if we ascertain what 
is the value of the term =+y (0,01)%, we may, by taking the hun- 
dredth part of this value, judge of the degree of exactness to 
which we shall have the value of the arc, if we confine ourselves 
to these four first terms. Now 54, (0,01)° is 


=z (0,000001) = ee = 0,0000000446, 


Fig. 46. 


108 Integral Calculus. 


of which the hundred part is 0,000 000 000 446; we may there- 
fore, with certainty, estimate each term of our series as far as to ten 
decimals, without fearing that the valueof the resulting arc should 
be faulty in the ninth decimal. Thus we shall have 
+4z (0,01)8 = 0,000 000 0446 ; .2, (0,01)? = 0,0000075000 ; 
a — 0,0016666666 ; 

the sum of the series will therefore be 

0,1 (1,0016742112), or 0,100167421, 
confining ourselves to 9 decimals, and we might with perfect safe- 
ty admit even the tenth. 
_ Such is the value of the arc whose versed sine is the hundredth 
part of the diameter. If, therefore, we knew how many times 
the number of degrees of this are is contained in 360°, we should, 
by multiplying their length by the number of times, have the 
approximate value of the circumference. But this we do not 
know. 

As we know (Trig. 18) that the sine of 30° is half the ra- 
dius, and as, knowing the sine of an arc, we may easily find 
its versed sine, we might calculate the versed sine of 30°, 
substitute it for a in the above series, and then, multiplying 
the result by 12, which is the number of times that 30" is contain- 
ed in 360°, we should have the approximate length of the cir- 
cumference. But as the series would be little convergent, so that 
we should have to calculate a great number of terms, in order to 
obtain an approximate value of the circumference, we shall point 
out another way, which will serve as a second example of the 
method of approximation. 

Draw the tangent AN’ (fig. 46) the secant C/N and the se- 
cant Cmn infinitely near to it; from the center C, and with the 
radius Cu¥, describe the infinitely small are Vr, which may be 
considered as perpendicular to Cn. The small triangle rn 
will be similar to the right-angled triangle C.4 n, because, besides 
the right angle, they have a common angle at n; it will be also 
similar to the triangle C.4.V, which differs infinitely little from 
C.4An; we have therefore 


CN: CA :: Nn: Mr 


now the similar sectors CW r, CM m, give 


_CAXNn. 
ne Ca ? 


Approximation. 109 


CNT COM on Ch ON OM ig — OF 
oi on 


Calling, therefore, 4N, 2; the radius C.A, a; we have 
Nn=dxand CVY=vya2 +22; 
the value of M m will thus become | 
2 2 
a —— ; that is, Mm = ieee ae 
therefore 
fMn=AM= fo 
This quantity cannot be exactly integrated. In order to inte- 
grate it by approximation, we must put it under the form 
Jf@ dx (a? +03); then, pe found (lg. 144) that 
(a7. os) = oT > (i—: ——+ oily at hah ++ = — &e.) 
Pe ay fig par ’ 
we shall have 


a x4 6 8 
Seda (a +a*)9= fde(1—54 ta ata &.) 


xidx x*dax r'dxa wxrdx 
=f (¢e—75 aa: calle pe ok ee ER —&e.) 
as as at x? 


=X — sa that Fast gas 


LS eS CO ne ee & 
Ema! fy 1 —— SECTS seraraet pry PIE SITET C. e 
( sai? Sat ash Oa® ) 


It now remains to ascertain whether we know any arc, which, 
being contained a known number of times in the circumference, 
has a known tangent. Now the arc of 45° is such an are, being 
contained 8 times in the circumference, and its tangent is equal 
to the radius ; supposing, therefore ~ = a, we shall have for the 
length of the arc of 45°, the value of this i 

a(l—24+}—344—4, &e,) 
But, as the terms of this series decrease very slowly, we must 
endeavour to decompose the arc of 45° into two other arcs, whose 
tangents shall be known. It is not important that the number of 
degrees in these arcs be known, provided they make 45°; when 
we shall have calculated their lengths by means of their tangents, 
we shall, by adding these lengths together, have that of the arc of 
45°, As these arcs will be less than 45°, their tangents will be 


110 Integral Calculus. 


less than the radius, and, consequently, the series will be more 
convergent and more easy to calculate. 

What has been laid down (Trig. 27) furnishes us the means of 
finding two such arcs. or, we found that a and 6 being any two 
arcs, we have 

tang a + tang b 
tang (a + 6) Yet oae tans b 

supposing the radius = 1. If, therefore, we suppose a +6 = 45°, 
in which case tang (a + 6) = 1, we shall have 

tanga -+tangb _ 

1—tangatangb” ’ 
an equation, from which, by the common rules, we deduce 
1 — tang a 
i+ tanga’ 
Taking then the tang a = 43 we shall have 


hs 

2 
i dae 
We have therefore only to calculate, by means of the series 


tang b= 


tang 6= =F. 


ene, 
above, the length of the are whose tangent & is g oF one half of 


via 
the radius, and the length of the arc whose tangent is 53 these two 
lengths added together will give the length of the arc of 45°, 
: . F ° a DA oe J 
Now, substituting successively Bi and 3 in place of & in the pre- 


ceding series, we have 
a 1 1 1 1 1 1 
AC —3 a2 15024 7.28 19,98 i1.gret gar —&- ) 


1 1 I 

(is “- arte a + 9,3? 11.310 t Jg;g12 kc. ) 

If we wish to have the value of each of these arcs expressed 
exactly as far as the ninth decimal, we must calculate the first 
15 terms of the first, and only the first LO terms of the second. 
Now this calculation is very easily made by observing that in the 
first, we may calculate the consecutive terms, by forming at first 
a series, each term of which shall be equal to the preceding mul- 


tiplied by ee that is, shall be + of the preceding; we then mul- 


tiply this series, term by term, by the series 1, 3, 4, 4, 4, &e. 


Approximation. {il 


finally, adding the odd terms together, and also the even terms to- 
gether, and subtracting the sum of the former from the sum of the 


: . a . 
latter, we multiply the remainder by oh In like manner, the cal- 


culation of the second is reduced to forming a series, each term of 
1 
32? 


shall be 3 of the preceding, and then proceeding as above, only 


which shall be equal to the preceding multiplied by —, that is, 


any a. a 
that the result is to be multiplied by = instead of : If this ope- 
ration is exccuted, and the approximation carried as far as 10 


decimals, we shall have for the first series “ (0.9272952180 : 
2 


or a (0.4636476090) ; and for the second, — (0.9652516632), 


or a (0.3217505544) ; therefore the arc of 45°, which is the sum 
of these two, will be a (0.7853981634). Taking, therefore, the 
quadruple, in order to obtain the semi-circumference, we shall 
have a (3.1415926536) ; therefore the radius is to the semi-cir- 
cumference, or the diameter is to the circumference 
2: @: a (3.1415926536) :: 1 : 3.1415926536, 

aratio which does not differ from that obtained in Geometry 
(art. 294), and which may be easily carried to still greater exact- 
ness. 

111. Asa third example of approximation, let it be proposed 
to find the logarithm of any number. But we must first observe 
what has been already mentioned (26), viz. that the logarithms 
here spoken of are not those found in the tables. But the former 
being calculated, we may immediately deduce from them the lat- 
ter, as will be seen as soon as we have shown how to calculate the 
former. 

We conceive the proposed number to be divided into two parts, 
represented by a+a; a being the greater part. According to 
what has been shown (26), we have 
dx 

d . log OY ho Lae rare 
a quantity which cannot be integrated algebraically. It must 
therefore be reduced to a series, and to this end put under the 
form ..da(a+2a)-'. Now we have (Alg. 144) 


112 Integral Calculus. 


a? as 


2 3 
(a4a)'=o'(1—2 45 © + be.) 
1 yb Os x3 
ACETIC sig 
whence 
dl(a+a)=dx (a+x)-! 
ie. Ae Ae xs dx 
yA Ger Ma) para, 
integrating, we have 


l@t)=C—-S+5- = +8) +0, 


2 at" Sas + or 
In order to determine the constant C, we observe that as this 
equation is universally true, it must be so when 2=0, in which 
case it is reduced tola =C’; therefore C =la, wherefore 


x xe xs 4 
I(a-a)=la+(~— S425 FS, + &e.) 
Knowing, therefore, the logarithm of a single number, we 
may, by this series, calculate the logarithm of any other number. 
If, for example, we suppose a=10, and a+-2x=11; we shall 


have «=1, and consequently ~ igs 39, Whence we shall find 


(0,1)? | (0,1)° 
; 2 — &e.) 


which shows what must be added to the logarithm of 10 to give 
that of 11. 

But, as the general series just found is not sufficiently conver- 
gent, we may proceed in the manner following. Let it be pro- 
posed to find the logarithm of a fraction whose numerator is 
greater than its denominator; we shall presently see that the in- 
vestigation of the logarithm of any number may always be refer- 
red to this case. 

Let a represent the sum of the numerator and denominator of 
this fraction, and « their difference ; then (4/g. 3) we have 
ta+ } for the ASL aud 3 a —43 & for the denominator ; 


am Ee 


and consequently = for the fraction ; or, suppressing the 


wv 
common factor 4, the fraction will be represented by = Got aa and 


consequently / sate) =/(a+a)—l(a—z) will represent its 


Logarithms. 113 


logarithm. If now we differentiate, considering a as constant, and 
x alone as variable,f we shall have (26) 
d x dx Qadx 


Pen ee a? — 42)-1, 
ata 'a—x at—ax? adu( a”) 


reducing, therefore, (a® —a?)—'to a anes (Alg. 144), we have 


8 
(a? a ed Coe hates = +5, + ke.) 5 
wherefore 
Be aw.) ne i". oS 
2ade (a —a)-1=2e Mat 5+ 54545 +h.) 
sci ik eS he), 


a? a? a? 


Qadx e atx 
oes Cd 0H log (= —x 
satin +s5+ 9 was + &e.) + os 
The value of the constant C may be determined as above, by 
examining what the equation becomes whenz=0. Now it is then 


= 
Therefore 


reduced tol = = C; therefore C =/1 ~ —11=03 we have 


therefore merely 
GT OS 
law 2(¢ tithe tra tia t &) 


in which we see that each term is formed from the preceding by 


multiplying by the square of 7 or of the first term, we then take 


the first term, + of the second, 4 of the third, &c., and double the 


sum. 
Let us apply it to some examples. Suppose, for instance, it 


t Though this fraction represents any proposed fraction, this does 

not hinder our considering the sum a of the numerator and denomi- 
“nator as constant, because there is no fraction which may not be so 
prepared as to render the sum of the numerator and denominator 
equal to any number whatever. For example, to bring the fraction 
3 to such a form as to have 12 for the sum of its numerator and de- 
nominator, we have only to multiply the two terms by 2, which gives 


—, and make 3u + 5u =12, or 8 u = 12, whence we have 


u= 12 =3; then } = =, of which the sum of the numerator and 


2 
oe 
15 

. . . * 
denominator is, in fact, 12 
15 


114 Integral Calculus. 


were required to find the logarithm of 2. In order to this we re- 
present 2 under the form 25; we shall then have a — 3, and x=1; 


x ee : 
wherefore -=4, and —-~=12. Each term may then be easily 
a a 


formed, as we have only to take 3 of the preceding term to form 
" 3 5 
° x x xv 

the series —, ——, 
Ti 8 


, &c. thus we shall have, 


— = 0,333333333 .. Therefore — = 0,333333333 


rey KO OPO SL die ete i Mes ose wie = 0,012345679 
x "te 
—— = 0,004115226 . —— = 0,000823045 
ae 5 a> 
at at 
— = 0,000457247 . —— = 0,000065321 
a? Zat 
x? a 
—— = 0,000050805 . . . . « ~~ =0,000005645 
a? 9a? 
— = 0,000005645 . . . . 1 0,000000513 
a lla 
713 xis 

= > =0,000000627. . . . ——~ = 0,000000046 
ai3 13 ai$ 
mis mis 
nae = 0,000000069 . . ME aegis = 0,000000004 


of which the SMS); 30 Wee OT Gado 
and double this, or the log, 2, is 0,693147176, which, confining our- 
selves to § decimals (for, to answer for the accuracy of the ninth, 
we ought to have carried the approximation farther) is 0,69314718. 

Since 4 is the square of 2, and 8 is its cube, double this loga- 
rithm will be the logarithm of 4, and the triple of it will be the 
logarithm of 8. 

In order to obtain that of 3, we may calculate, in the same way, 
the logarithm of the fraction 4, which being taken from that of 
4, will give the logarithm of 3, since 3 1s 4 divided by 4, there- 
fore 13 =14—14; but it may be found more easily by calcu- 
lating the logarithm of the fraction 8, and subtracting it from 
the logarithm of 8, which has already been found; the re- 
mainder will be the logarithm of 9, half of which will be the 
logarithm of 3. Adding that of 3 to that of 2, we shall 
have the logarithm of 6. In order to obtain the logarithm of 5, 
we must first find that of 10, by calculating that of 4°, which, 


Logarithms, 115 


being added to the logarithm of 8, will give that of 10.. By sub- 
tracting from this last, the logarithm of 2, we obtain the loga- 
rithm of 5. 

We thus see what is to be done in order to calculate any other 
logarithm. But it ought to be observed, that the calculation be- 
comes shorter and shorter in proportion as the number becomes 
greater, so that when we have the logarithms only as far as 10, 
we may calculate. the others as far as 100 without employing 
more than three terms of the series, if we confine ourselves to 8 
decimals ; beyond 100, the two first terms are sufficient, until we 
get to 1,000, and above that, a single term is sufficient. 

In order to reduce these logarithms to those of the com- 
mon tables, we must previously have the logarithm of 10. Now, 
if we calculate the logarithm of 4°, by the preceding formula, 
we shall find J 1° = 0,22314355; adding to this the lc garithm of 
8, which is found by tripling the logarithm of 2 obtained above, 
we have / 10 = 2,30258509. 

112. Having done this, we observe that the equation 

d c= yeas 
y 
on which (26) is founded the present calculation of logarithms, 
agrees only to the system of logarithms in which we suppose the 
modulus = 13; but that the equation which applies to all possible 
systems of logarithms, is 
mad 

d.0 = Sree 
and that which applies to all the’systems of logarithms in which 
we suppose that the first term a of the fundamental geometrical 
progression is 1, is 


The first, viz. dx = a gives, on being integrated, w=ly; 


m da 


and the second, viz. dx = , gives = ml y, which shows, 


since wrepresents the logarithm, that to reduce the logarithms re- 
sulting immediately from calculation to those of any other system 
whose modulus is m, we must multiply them hy m. Now the 
logarithm of 10, in the common tables, is 1; and we have just seen 
that the logarithm of 10, which is given by the calculation, is 


116 Integral Calculus. 


2,302558509 ; we have therefore m x 2,30258509 = 13; where- 


1 . 
fore the mod les is ———-—— 
ulus of the common tables is 550258509" which, by 


performing the division, is reduced to 0,43429448. 

Therefore, to deduce the logarithms given immediately by the 
calculus to the logarithms of the tables, we must multiply them by 
0,43429448. And, reciprocally, to reduce the logarithms of the 
tables to those resulting immediately from calculation, we must 
divide them by 0,43429448, or, which is more convenient and comes 
to the same thing, multiply them by 2,30258509. 

Thus, if 0,69314718, which was obtained above as the loga- 
rithm of 2,be multiplied by 0,43429448, it gives 0,3010300 for 
the logarithm of 2, which it is in fact in the common tables. 


113, If we wish to go back from the logarithm to the number itself, 
we may proceed thus. We have seen above, that, representing any 
number by a + x, we had 

es 


x cat a4 


therefore 


at+ox x x2 Sd fo 
VO ik) ey iS. iia SpA BR FE 82) 
AO ata | a a 202 Sa% 444 i 


a being an arbitrary number, but such that its logarithm may differ 
little from that which is given, and which is supposed to be the loga- 
rithm of a+. For the sake of greater simplicity, let 


1 € an *) ue a 
‘a 
and we shall have 


x x2 i x4 
a 2a? 8 a3 4 a* 


a 


It is required to obtain the value of ~ in terms of ~. 
Let us suppose that this value may be expressed by 
~ =Az+ Bz? +0z3+ Dz + &e. 


4, B, C, &c. being constant coefficients, which we wish to determine. 
We shall therefore have 


; 


Logarithms. 117 


. = Ax ag Bzr7+ Cz + Dz 4 &e. 


ae? A? 2AB Sad st. 
mae ee I 3 mm 4 ee OC, 
ye a = = % &c 

2AC 
z or ° ° — ——-<z* — & 

2 

3 AS 3 A? B 
—_—_— = e ° e . a e ns aay 3 } 4 & 
T Fas 3 heat 

Od A+ 
—_—_ =< ° e e ° e e e e barge 4 cr aeee 4 bea | ° 
ee 4 Z &c 


Now, in order that this equation may hold true, whatever be the 
value of z, it is necessary, Ist, that 4 = 1; 2d, that the sum of the 
terms which multiply each power of z in the other columns be zero. 
We have therefore, 


2 3 
Boe ce oN Vay a cl 
2 3 


% A 


J 4 
whence we deduce 
I 1 1 
us Say g belly pal Neamt eB een er ey ans 
ff pa, ae ae Nyssa 3s 13933 54:¢ 


and if we supposed a still greater number of terms in the series, 
such as Ez, Fx, &c. we should find, in like manner, 


Ratha 


we have therefore 


2 a 
pire le a 


Therefore 


23 xz we Ag 


[2.5.47 ¥ 


12.345 1 12.3456 1 &*: 


x at ow 
2h oe ae 
2 x? zoe x4 og 

=P gar Teaninra SM ers arate Seale +X 

To make use of this formula, we must take from the given loga- 
rithm, which is that of a + «, the nearest known logarithm, the num- 
ber corresponding to which we take for a Then we shall have 
a+ x 


t , or x) which may be substituted in the preceding formula. 


ate 
a 


The result will be the value of 3 whence we may easily de- 


duce the value of a + a, since a will be known. 

As we speak here of the logarithms which have 1 for the modulus, 
if the logarithm given were of the nature of those in the common 
tables, we should have to begin by reducing it, and also that which 


118 . Integral Calculus. 


we took for the logarithm of a, or merely their difference, to these log- 
arithms, which might be done by the methods given (112). 
If we wish to know what is the number whose logarithm is 1, in the 


system here treated of, we must suppose i(“=*), or z= 1, and 


we shall have 


atx 1 1 1 1 
peel tae th ee tye ata aes alee 


which will be found = 2,7182818, confining ourselves to 7 decimals. 


The number whose logarithm is 1 is often met with in caleula- 
tions ; we shall meet with it hereafter, and it is for that reason that 
we have shown the method of calculating it. 

114, We may have another expression for a number by means of 
its logarithm. If, for example, x were the number given, and 
Lx =, we multiply the second number of this equation by le, e be- 
ing the number whose logarithm is 1, and have l « = x le, which 
does not effect any change, since le =1. Now the equation 
Lx =z leis changed, by the nature of logarithms, intol x2 = le; 
whence we deduce x = e*, because, the logarithms being equal, the 
quantities to which they belong must be equal. 

But, agreeably to what has been said in article 113, if we have 


bx =z,wehavexr=14-x%+ i = + &c. Andas we have at the 


same time, 7 = e*, we shall have 


id as ze 
Ae ae ters Tye Laan 
Remark. 
115. The method which we have employed to deduce the value of 
2 2 f \ 
x from the equation z = - — = + &c. is called the inverse method 


of series. It consists, as we have seen, in supposing the variable, 
whose value we would find, to be expressed by a series, in which the 
other variable has exponents in arithmetical progression, and each 
term an indeterminate constant coeflicient. 

If we had several terms in 2 and z in the same equation, without 
having « and z multiplied together, we might determine the series of 
the exponents, by making the exponent of the first term of the sup- 
posed series, equal to the smallest exponent of the same variable in 
the given equation, and taking as common difference of the exponents 
of the same series, the greatest common divisor of the exponents of 
this same variable in the equation, If we had, for example, 


2 
z3-+3% = 2H —2774 305+ &e., 
we should make . 
2 A 5 
xe=AB+B24+E24+Dz54+ Ex? + &e. i 
because the least exponent of z is 2, and the greatest common divisor 


c -_ 2 
of the exponents 2 and 1, of z, is 4. 


Integrating by Circular Arcs. 419 


But if « and z were multiplied together, it would be necessary to 
pursue a method, the detail of which does not fall within the limits 
of our plan. 


Uses of the preceding Approximations, in the integration of 
different quantities. 


116. As there are tables already calculated of the different 
parts of the circle, as well as of logarithms, when we have to in- 
tegrate any differential, which may be referred to the circle, or to 
logarithms, it will hereafter be unnecessary to reduce these dif- 
ferentials to series, as we may refer at once to those tables. We 
shall therefore now point out such of these differentials as most 
frequently occur, and show how the arcs of a circle, or the loga- 
rithms, which are their integral, may’ be determined. 


117. We have seen (98) that te Ge expresses the element 
/ax—x2 


of a circular arc 4M ( fig. 40), of which a is the diameter and « Fig. 40. 
Zadx 


the absciss ; so that the integral of this quantity, or f <> 
L— @ 
is represented by the arc 4M. Suppose, therefore, that it is re- 
quired to find the value of this integral for a determinate value of 
x; from C4 or 1a, we subtract the known value of x or AP, 
and obtain CP. In the right angled triangle CPM, we know the 
right angle, the hypothenuse C/M = 2 a, and the side CP; we 
may therefore calculate the angle 4C.M; and knowing the angle 
ACM or the number of degrees in the arc 1M, and also its radius 
CM, it is easy to calculate the length of this arc (Geom. 294). 


118. If we had hed © 


gkxr—pxe 
quantities, we should render this differential similar to the pre- 
ceding, by dividing both terms of the fraction by 4/p, which would 
give 


, A, g, p, and k being known 


now if we had, as multiplier of d x, half of the quantity g te by 
P 


which x is multiplied in the radical, then this differential would 
be similar to that of the preceding article, we therefore give it 


120 Integral Calculus. 


that condition, by multiplying and dividing, at the same time, by 


k gk 
3. = or a we shall have 
h ok gk 
va see nih 3p 


When in el ea we see that the integral of our differential is 
an arc of a circle, whose diameter is oe and absciss #, it 1s, we 


observe, such an arc multiplied by if as ; it is therefore easy 
Sk Vp 


to be assigned by the method just given. 

119. If, instead of reckoning the abscisses from the point 2, 
we had reckoned them from the centre C, calling the radius CA, 
f —iadx 
4d a,and the absciss CP, x; we should have had ————— 

qa4a—w7 & 
for the element of the arc AM; which is easily found by com- 
paring the similar triangles CPM, Mr m, remembering that 
PM =/302 —x2,; 
and that, inasmuch as the arc 4M diminishes in proportion as 
CP or @ increases, its differential must be negative. ‘Thus, when 


kda f 
we have such a differential as —————_,, we may change it as 
Vgh—pxe 
dx 


i k 
above into —. —_—=::—== 
Vp Ne h 
2 
Pp 


the quantity — 2 a, which ought to be in the numerator, is— ee 


gh 
; how =— representing here 4a a, 
v 


we therefore both multiply and divide by — Jee hand have 


Supposing, therefore, that C.4 = Re and CP, x, we shall 


have 


Integrating by Circular Ares. 121 
a 


a 


x AM, for the integral, or ett generally 


Bem rea Cat ans C. 


oO 
2) 


P 
As to the constant C, it is determined by the conditions of the 


particular question, which may have led to the differential under 
consideration, and the arc 4/M is determined, as we have just 
seen (art. 118), that is, by the calculation of the triangle CPM. 
anadx 

re atae c 
circle, of which a is radius and av tangent, an are which may be 

easily determined for any determinate value of 2, by calculating 

the angle 4CN of the right-angled triangle CN ( fig. 46), and Fig. 46. 
then the length of the arc 4M by means of the radius a and of the 
number of degrees in the angle 2CWN. 


120. We have seen (110), that expresses an arc of a 


If, therefore, we had renee we should divide both terms 
by h; which would give = x —““ —, then multiplyi 
yh; which would give — oar en multiplying both 
ae 
terms by 2 gue =, we should have 
ke g b2 g b2 
CaM Ecole ie hod? 
SRS NERS.) Tle AR eh TAL 
a oS tee $0 ET tes 


we should therefore have the integral by calculating the length of 


the arc which has 2 for its tangent, and ee for its radius ; 


and multiplying by ap 
121. These three differentials are therefore integrated by ares 
of acircle. The following are integrated by means of the surface 
of a circle. 
The element of the half segment 4PM (fig. 40) is d @ V/az—z x, Fig. 40, 
ealling 4P, x; since y=s/ax—xzx, and consequently yd x or 
16 


Big. 47. 


Fig. 40. 


122 Integral Calculus. 


PpmWM=dx /axz—xz; therefore every differential having 
this form, or susceptible of being reduced to this by preparations 
similar to those just pointed out, may be integrated by means of 
the half segment of a circle, whose absciss is # and diameter a, a 
segment easily determined as well by what has just been said, as 
by the methods of elementary Geometry. 


If, for example, we wish to find the surface of the ellipti- 
cal half segment 4/MP (fig. 47), we have y= “ /ax— 2% 5 
wherefore 

ydoxd (APM) =-"% , Yas—a3} 


now dX ./ax—x-x expresses the element of the circular half 
segment 4PM’, supposing a circle to be described upon 4B, asa 
diameter; we have therefore 


d (APM) = d (APM), 


and integrating, 4PM = z APM’, which gives 


APM. + APMS:': 8: as 
that is, the surface of the elliptical half segment, is to the surface 
of the corresponding circular half segment, as the less axis is to 
the greater axis; whence it is easy to conclude that the whole 
surface of the ellipse is to that of the circle described on its 
greater axis, as the less axis is to the greater ; which we promised 
(107) to demonstrate. 

If, instead of reckoning the abscisses from the point 4, ( fig.40), 
we reckon them from the centre C, then calling €.4, 4a, and 
CP, «x; we shall have —da4/daa—<az for the element of the 
half segment 4PM; because then y= 4 d7a—~ra, and the seg- 
ment 4PM diminishes while z increases, which makes the diffe- 
rential of 4PM negative. 

The following is an example of a differential, which refers itself 
to this form. 

122. Let it be proposed to find the surface of the elongated 
elliptical spheroid. ‘The general formula for this kind of surfaces 


is =! s/da2 + dy2 (98) ; now the equation of the ellipse is 


2 b? 1 2 2 
y Ti 8 (4 a? —x?); 


Integrating by the Surface of a Circle. 123 


therefore 
BY cee b xwdax 
=_— — a d d =e — pieces Lea AY ' 
DN eet ada ae 
therefore 


c prides eas 4 B14 Pl nea 

tian +dy2 => — me gates x faa x, 
an +a* — x® 

or, performing the oa indicated, reducing, and raising 

dx? from under the radical, 


chdx pe Cchd writs oad 72 hee 
a aa ra che aware mR eT TE 


ra aa 


now, if we call the distance CF to the focus F, k, (fig. 48), we Fig. 48. 

have k*? =1a? —163, or 4k? = a? — b? (Trig. 112); there- 

fore the mene of the surface becomes 

tee chdax pas 4 ka as. 
ra aa raa 

Dividing under the radical by 442, and multiplying out of it by 


? 2chkdx tat 
its root 2k, and we have ———_—- kab — x, a quantity to 


rad kk 
which we must give the sign — to make it express the surface 
reckoned from the point 4, because this surface diminishes in 
proportion as x increases ; thus we have 


_ 2chkda 1 a4 : 

raa Pa 
Comparing this with — dw 4/02 —z2, which we have just found 
to be the expression for a circular half segment, whose radius is 


1 a, we shall conclude that the integral of —d x Jz Oe nat ie 
kk 


a circular half segment OM'P, whose radius is a f and whose 
absciss taken from the centre is 2, plus a constant quantity. 
Therefore, if with a radius CO= ct that is, a third proportion- 
al to CF and C4, we describe the circle OMR, we shall have 


/- da free as, *— 2x? = OPM +C; 
therefore 


Jar Sete opm + 2°28 x ¢, 
. rad rad 


rad 


124 Integral Calculus. 


In order to determine the constant C, we must observe, that 
the surface sought having its origin at the point 4, must be zero 
at that point; but, at the point 4, the half segment OPJW' be- 
comes O.4.N’; we have, therefore, 


_2 chk x OAN 4 Qch b, 
whence we deduce C =— O.A.N’; oes the entire integral is 
oct x OPM — *5*"x OAN, or =< (OPM'— O4N), 
or finally 
200% (APM'N). 
Therefore the surface of the half spheroid will be 
pi cea oh 08) 
or, since CO = a , and apennn! aa 3 SOO this surface 
will be x 57 x ACR, ot © x FO. x ACRN'3 and that of 


the sinalé spheroid is double of U ne 

The mode of determining the radius CO is very simple. From 
the point C, as a centre, and with the radius C4, we describe the 
arc AL, cutting in L, the line FL perpendicular to CA from the 
point F’'; we produce CL until it meets at JV the perpendicular 
AN drawn seks the point 4; this gives CV for the value sought 


of CO, ; for, the similar triangles CFL and CAN, give 
Coe Ges! CA es CN, 
1 i a? 

or fe a Site. = 


We may thake use of the above result in measuring the surface of 
the hull of ships, which may be compared to the surface of an ellip- 
soid even more properly than their solidity can to the solidity of the 
same ellipsoid. ‘he whole operation consists in calculating the an- 
gle ACN of the right-angled triangle CNV, of which we know two 
sides and the right angle. Thus the angle WCR becomes known, 
and we may easily deduce from it the surface of the sector VCR, 
to which adding that of the triangle CAV, we have ACRN, 
which we have only to multiply by — x —. When we have 
found the surface of the hull, by Eee ing it by the thickness 


Integrating by Logarithms. 125 


ef the sheathing, we obtain the solid contents of the volume by 
which the hull is increased by sheathing. 

123. With regard to the quantities which are immediately re- 
ferred to logarithms, they are those in which the proposed differ- 
ential is, or may be made to be, a fraction, whose numerator is 
the differential of the denominator, or this differential multiplied 
or divided by a constant number. 

When the numerator is exactly the differential of the denomi- 
nator, the integral is the logarithm of the denominator. ‘Thus 


Sales; [PA =lata) tes 


Qxedx 
ee a =I (a? +27) +C. 


But when the numerator is the differential of the denominator, 
multiplied or divided by a constant number, the proposed differ- 
ential must be decomposed into two factors, of which one shall 
be a fraction having for its numerator the exact differential of the 
denominator, and the other a constant number. Then the inte- 
gral will be the logarithm of the variable denominator, multiplied 
axtdz 
pers aes 
the differential of a + 2° is 3x27 da, the differential must be so 
prepared as to have 3x7 da, in the numerator. To this end, we 
write it thus: 

er Bice) hoe 
3° a8 +23 
In like manner 


A = {= eee =—l(a—a2)+C 


=0—I(a—a)+C=li—1(a—a2)4C=11_1¢. 
a—wDH 
So 


sda Qxd ———_ 
Nee teak rc +22)+C=l fata 40. 


Finally, 
ax®—'dax = f=. woe? dic 
k++ ba" bm k++ ba" 


=qol(k+bar)+ Cal (e+b anyon + C. 


by the constant factor. For example, to integrate 


: - a 
, whose integral is at (a§ +25) 4 C. 


126 Integral Calculus. 


The following is an example of the manner of determining 
these integrals, in numbers. Suppose that it is required to find 
the value of /(a+~2), (a being 5,) when a is Z. [tis then 17 
which we wish to find. We take the logarithm of 7 from the 
common tables, which is 0,8450980; we then multiply it (112) 
by 2,30258509, or 2,3025851, and obtain 1,9459100 or 1,94591 
dx 


,whena=5 
a x 


for the value of / (a + 2) or of the integral of 


.and = 2, 
We sometimes meet with differentials which are integrated direct- 
ly by logarithms, although they cannot be prepared like the preced- 


ing. For example, is of such a kind. We sometimes 


dx 
/xrx—i 
succeed in giving them the form of a logarithmic differential, by mul- 
tiplying them by such a function of « that the product may become 
the differential of this function, or that differential multiplied or di- 
vided by a constant number. If then we divide by this function, the 
differential would evidently become a logarithmic differential. Ap- 


ai we multiply it by « + x3 —1, 

3 et « 

and we have SS 2S + dx, which is the differential of a + ./73—7; 
x — 


so that we have 


plying this observation to 


d xdx 
dix iat eae 


acl She SIO SI elt dd ildcnhadear ta daisies MY a | 
Wenn ase Eada (2 + ./ 7x38 i) + C. 


d. 
i ipaerit by first multiplying 


both terms by ./—1, which gives a 
have just seen, is ./—1 U(x + /za—1) + C. 


We shall also find the integral of 


» whose integral, as we 


124. In art. 82 we promised to explain how it happens that 
ihe fundamental rule for the integration of simple quantities gives 


an infinite quantity for the integral of ne while this integral is 
expressed by / a, or at least la + C. 

The integral ate may be finite or infinite, according to the 
portion of it which we choose to take. To illustrate this, we 


dx’: ; 
first observe that to take the integral of serge nothing else than 


Reduced Maps. 127 


to square the common hyperbola, considered with reference to its 
asymptotes. For the equation of this curve is 

xy=aa, or Xy=1, 
if we suppose, for the sake of greater simplicity, thata = 1. Now, 


: P 1 
from this equation we deduce y = Pa wherefore yd, the ele- 


ment of the surface, becomes to ; if therefore we wish to have 
the space reckoned from the asymptote 4Z (fig. 49), the inte- Fig. 49. 
gral of ard or 1 x + C must become zero, when the point P falls 

x 


upon the point 4, or when x =0; in which case, therefore, we 
have /0 + C= 0, and consequently C= —/ 0 ; therefore the inte- 


gralis la—i 0 orf = 5 that is, the space ZAPMO reckoned from 


the asymptote is infinite, Z and V being considered as the ex- 
tremities of the asymptote and of the corresponding branch of 
the hyperbola; in which there is nothing surprising. 

But if, the point O being the vertex of the hyperbola, in which 
case the corresponding absciss 4/V=1, we wish to find the 
space reckoned from the point JV, the integral 1x + C must be- 
come zero, when the point P shall fall upon the point JV, or when 
x2 =1; we have therefore /1-+ C= 0, and consequently 

C=—l1=0. 
wherefore the space VOMP is expressed by ta. 

We see from this, Ist, that the logarithms immediately result- 
ing from calculation, express the hyperbolic spaces comprehend- 
ed between the asymptote of the curve, and reckoned from O 


d 
the vertex of the curve ; 2d, that if the integral S* or a-tdax 
x 


is infinite when found by the fundamental rule, it is because it 
expresses the space reckoned from the origin of the asymptotes. 

125. As an application of integration by logarithms, let it be 
proposed to explain and apply the principles of the construction 
of reduced Maps.+ 

These maps have been invented in order to facilitate the laying 
down courses in navigation. ‘The course, at least for a certain 
time, is constantly on the same rhumb, and consequently makes 


a 


+ Maps on Mercator’s projection. 


Fig. 50. 


Fig. 51. 


128 Integral Calculus. 


constantly the same angle with each meridian which is succes- 
sively crossed. Whence it follows, that if we wished to trace 
this course on a common map, in which all the meridians tend to- 
wards the same point, we should find it a curve line, and conse- 
quently very inconvenient for the operations which it is necessa- 
ry to perform. Ithas been found better to represent the meridi- 
ans by parallel straight lines, in order that the course which 
makes a constant angle with these meridians might be a straight 
line. 

But, supposing that 1MP, am P, (fig. 50), are two meridians; 
Ala, a portion of the equator comprehended between these two 
meridians, and Mm the corresponding portion of any parallel ; 
we see that the interval WM’ m’, ( fig.51), which represents the arc 
M m, is of the same magnitude as 4’ a’, which represents 4a, 
the corresponding portion of the equator ; although M m is smaller 
than .4 a, in the proportion of P'\M to C.1; which may be easily 
shown by drawing MP’, m P’, AC, and a C, perpendicular to CP, 
for these lines form with the arcs 4aand Mm, the similar sectors 
CAa and PMm. 

To compensate for the increase given to Mm, by representing 
it on the map by M’m’, we represent the latitude 4M by a line 
AM’, which exceeds 4’ a’ more than 4.M exceeds 4 a, in a cer- 
tain ratio. And it is this ratio which it is now proposed to ascer- 
tain. 

Let therefore M and R be two points infinitely near each other 
upon the meridian 4M ; Mm, Rr the corresponding portions of 
two parallels. If we wish to represent Mm, Rr by the lines 
M' m', R’r', equal to the line 4’ a’, which represents 4 a, and still 
preserve the same ratio between the parts of each parallel, and 
those of the meridian, we must make the infinitely small interval 
M'R', which separates M m’ and R’7', as much larger than MR 
as Mm in smaller than 4 a, that is, we must have 

MR’: MR::Aa:Mm:: CA: P’M:: as the radius is 

to the cosine of the latitude 4M. 

If, therefore, we call the latitude 2M, s, the straight line #’.M, 
which represents it on the map, and which is called the Increasing 
latitude, will be indicated by s’. Take the radius C4 or CM= 1; 
CP’ or the sine of the latitude 4M = x3 we shall have 


Reduced Maps. 129 


: ae d x 
PM =z Vi 23 5 nee ae ; 
P'’M:CM:: tR: MR, or ViI-z2 :1:: dz: ie: 
/1— «2 
But PM:CA:: MR: MR; 
ait a dx Awan eee 
therefore: /1—x2 s Maer Rarer Tass eee 


It is therefore necessary, in order to obtain s’, to integrate the 


value of ds’, that is, set Now, we have seen (111) that 


Qadx x 
3 —— resulted from the differentiation of pi alee ; therefore 
a* — 2X Qa 1 


d Ls 
— results from that of ie oa eit and consequently the integral 
wv 


of ——, which is one half of © = ‘will be 31 sas ac =, we 


shall therefore have 


an integral to which there is no constant quantity to be added, 


because when x = 0, 1 fe +” becomes log vi =l.1=0; now * 


s', or the straight line 4’M’, must in fact be in that case ZeYO, 
heditice the arc AM, etch it represents, is zero when @ is zero. 

It may be now observed, in order to render this expression 
more convenient, that the radius is the sine of 90°; and since by 
« we have understood CP’, or the sine of the latitude AM, we 


shall have, instead of s’ = =! ite ed , this equation, 


ia log, fo 90° + sin AM, 
sin SU°— sin AM 
Now, (Trig. 28), 
sin 90° + sin AM tang (45° + 4.4M) 
sin 90° — sin Avid ~ tang (45° — 2 AM)? 


therefore 


s' = log tang. vee ure AM). 
tang .(45°—24M) 


But (Trig. 9), (tang 45° — 24M) :1::1: cot (45°—21 4M). 
17 


130 Integral Calculus. 


Moreover, cot (45° — 4 4.¥) = tang (45° + 1 AM ), because 
45° — 3 AM is the complement of 45° + 1 .4.M, since 

45° — 1 AM= 90°— 45° — 1 4M; 
we have therefore 

tang (45°-— 5 1M): 1:: 1: tang (45°+3 AM); 
therefore | 
1 
tang 25° + 3 AMY 
Substituting in the expression for s’, we have 
s’ = log tang (45° + 4.4M)2 = log tang (45° +1 4M) 
= log. cot (45°—4.4.M). 
Now 45° — 1 4.M = half of 90°— AM, which is the complement 
of the latitude; we have therefore the increasing latitude 
s’ = log. cut (¢ the complement of the latitude). 

We therefore take, in the common tables, the logarithm of the 
cotangent of half the complement of the latitude, and, having mul- 
tiplied it (112) by 2.30258509, the product will be the increasing 
latitude expressed in parts of the radius. 

But, as it is more convenient to have the increasing latitude 
expressed in degrees, it may be thus determined. We found 
- (110), that the length of the semicircumference of a circle, whose 
radius is 1, is 3.1415926, &c. ‘This number, divided by 180, gives 
0.0174533 for the length of adegree. We have therefore only 
to find how many times this number is contained in the increasing 
latitude which has just been determined, that is, to divide the in- 
creasing latitude by 0.0174535 ; the quotient will express the 
increasing latitude in degrees. We obtain therefore the increasing 
latitude in degrees, by the formula 

2.30258509 x log. cot (4 co. latitude) 
0.0174533 

But, by dividing 2.30258509 by 0.0174533, we obtain, as quo- 
tient, 131.9283 if we confine ourselves to four decimals. There- 
fore, to obtain, in degrees, the increasing latitude, we must take 
from the common tables the logarithm of the cotangent of half the 
complement of the latitude, and multiply it by 131.9283. The 
product will be the number of degrees and parts of a degree of 
the increasing latitude. 

If, for example, we wish to find the increasing latitude corres- 
ponding te 40° of simple latitude, we take half the complement of 


tang (45° 14M) = 


Reduced Maps. ‘ 181 


40°, that is, half of 50° or 25°, The logarithm of the cotangent 
of 25° is 0.3313275,+ which, being multiplied by 131.9283, gives 
43.7115, that is, 43°.7115, or 43° 43’. It is thus we may calcu- 
late tables of increasing latitudes. 

126. It is by a calculation nearly similar, that we determine 
the difference of longitude, when we know the difference of lati- 
tude and the course. ‘The following is the manner in whiclrit is 
obtained. 

Let OQ. ( fig. 52) be the course, Q the point of departure ; 
OA the equator; AMP and amp, two meridians infinitely near 
each other. If we imagine the arc .@ m parallel to the equator, 
the infinitely small triangle m Mr will be rectilineal and right- 
angled at m ; we shall consequently have 

I:tangmrM::mr:Mm=mr.tang mri, | 
supposing the radius=1. If, now, we compare figure 52 with 
figure 50, we have, agreeably to what we have already seen in the 
preceding question, Mm: 4Aa:: cos lat: 1, that is, 

mr x tangmrM: 4a::cos lat: 1. 

If we call the sine of the latitude 4.M, x, its cosine will be 
Y1i—ax. The arc mr, which is the difference of the two lati- 
tudes ar and AM will be the differential of the arc 4.M, and will 
a (125). If we represent, more- 
over, by a, the angle Mrm, which the course makes with the 
meridian, or the rhumb, and by z, the difference of longitude 
BA, which give 4a =dz ; the proportion just found is changed 
into 


have for its expression 


Hii a tanca.dax dx 
I—avr:1l:: —2——:dz=tanga : 
6 xy Vg ae Fr) 1—X2x 
But we have just seen, in the preceding question, that the integral 
d 
of ——" was 1 cts we shall therefore have 
1—axe 1—z 
1. 
z= tanga by eallate 


To determine the constant C, we observe that when z= 0, that 
is, at the point of departure, the latitude 4.M becomes the latitude 


t As we have supposed the radius equal to unity, the logarithms of 
tangents must be diminished by the logarithm of the radius, which is 
10.000000 ; that is, we must take away the characteristic 10. 


Fig. 52, 


132 Integral Calculus. 


of departure BQ. Let t represent the sine of this latitude. The 
constant C, thea, must be such, that, substituting ¢ for a, we may 
have z=0. We have therefore 
1+¢ 
O=}tanga.l + C, 
and consequently 


C=—ttanga.l ae, 


Wherefore 


= tanga (gle —3 =) 


= tanga (1 BE Ny —). 
L— wv 1—t 
By reasoning precisely as in the preceding question, we 


shall find eae is reduced to cot (2 complement of 4.7); and 


fl Fe. 
for the same reason mE 7 7, 1s reduced to cot (1 complement of 


BQ), wherefore, the difference of longitude or z =tang a. (log. 
cot (3 co. lat. of point arrived at) — log. cot. (} co. lat. of point 
of departure) ; which furnishes a very simple rule for finding the 
difference of longitude, either by the tables of increasing latitudes, 
or by the reduced maps, whatever may be the course. 


By the Table of Increasing Latitudes. 


Find the increasing latitudes corresponding to the latitude of the 
point arrwed at, and of the point of departure. Take the diffe- 
rence of the latitudes thus found, (or their sum, if the latitudes are 
of different denominations), and multiply rt by the tangent of the 
course ; you will have the difference of longitude in degrees, min- 
utes, and seconds. 

By the Reduced Maps. 

Find upon the meridian the latitude of the point arrived at, and 
that of the point of departure. Draw, through the extremity of 
each, a perpendicular meeting the course proposed. The difference 
of these two perpendiculars, applied to the scale of longitude, will 
gwe you, in degrees, minutes, &c. the difference of longitude. 


Loxodromic Curve. 133 


ludeed, 4Q, (fig. 53) being the meridian; OL the course ; Fig. 53. 
AQ, AP the two increasing latitudes; PQ or RT is their diffe- 
rence; now, in the right-angled triangle SRT, we have 

Detained sus :.2 Fedcis 
therefore TS=RT x tang TRS ; 
now #T is the difference of the two increasing latitudes, and the 
angle T'RS is equal to the angle made by the course with the 
neridian. 

The curve line QM (fig. 52), which marks the course of the vig. 59. 
vessel on the surface of the globe, is called the Loxodromic. 

In whatever part of the course we suppose the point M, the 
triangle Mr m has always the same angles, since the angle mr M 
is always the same, and the angle m isa right angle. There is 
therefore always the same ratio between Jr and mr, as between 
the radius and cosine of the angle Mr m, which we have called 
a. Therefore the number of leagues in the distance QJM, is to 
the number of leagues made in latitude, that is, the number of 
leagues in JJM, as radius is to the cosine of the course. ‘This 
serves to determine the difference of latitude, when we know the 
course and distance in leagues. The same proportion serves to 
determine the course, when we known the difference of latitude 
and the distance in leagues. It is to be understood that the de- 
grees in the difference of. latitude are to be reduced to leagues in 
the proportion of 20 leagues to a degree. 


On the manner of reducing, when it rs possible, the integration of 
a proposed differential, to that of a known differential, and dis- 
tinguishing in what cases this may be done. 


128. We shall explain only the method for binomial differentials ; 
it will afterwards be easy to apply the principles to more complex 
differentials. 

Let us suppose, at first, that the proposed differential is 

hadz(a+ bx je, 
and that «™ d x(a + ba")? is that on which it is to depend, (or that, 
to the form of which we would reduce it); that is, let us suppose 
that the two exponents of the binomial are the same. 

We shall suppose 
fie da (a+ba")e = (a+ bar)Pti (Ack + Bakta4Cxkt+29 


MEAs: ere P xk +t) + f Ram da(a + b xn )e ; 


134 Integral Calculus. 


k and q being unknown exponents; ¢ a positive whole number; 4, 
B, C, P, R, &c. constant exponents, also unknown. Differentiating, 
we have 
hasda(a+bar\ 

=(p+1)nbar—dx(a+ ban) (Mak 4+ Bakto + Catt . 
+ Paktt?) +(atbar)jeti(ke Aavk—-1da+t(k+q) Bakti-tdax 
+ (k +29) Cxkt87-1dae. bk + tg) Patiq—-) dz) 
+ Ra ™dx(a+bar)P, 
or, dividing the whole by (a + ba)? dz, we have 
hat =(p+i)nba%—-\(Aak + Bakto +Cakt2a ....4 Pakt'7) 
+ (a+ ba") (k Axk-14 (ke +9) Bakti—-! +(k +29) Cakt2a-1 

slevels a ote Ue to b0) ete Gee tee leet 

In order that this equation may be still true, independently of any 
value of 2, it is necessary, that after the multiplications and transpo- 
sitions are performed, the sum of the quantities which multiply the 
same power of x be zero; it is by this condition that we determine 
the coefficients .2, B, C, &c. But, that this may take place, the num- 
ber of powers of «, which enter into this equation, must not exceed 
the number of these coefficients. | 

Now the number of coefficients, as may be easily seen, is t + 23; 
let us therefore find the number of the powers of x. Inorder to that, 
we must determine & and q. 

They may be determined thus. k—1 is the least indeterminate 
exponent found in the equation; we make it equal to m or to s, ac- 
cording as m ors is the least exponent. The greatest determinate 
exponent to be found in the equation is, as may be easily seen, 
k+tq-+n—1; we make this equal to s, if we have made 

k—-1l=m; 
or to m, if we have made kK —1 =s. 

Let us suppose & — 1 = m, we shall therefore have 

k+iqtn—l=s. 
This done, that the equation may not contain more powers of x, than 
there are indeterminate coefficients, the coefficients of «x in this 
equation must form an arithmetical progression whose difference is 
q, which cannot fail to be the case, since we have supposed 
k—1=m, k+ttg+n—l1=s, 

and ¢ a positive whole number. Now the greatest term of this pro- 
gression being kK + tq +n—1, and the least, k — 1, we easily find 
(lg. 230) the number of terms of this progression to be 


k+tqt+tn—1—k+1 fy 5 Baap 
Y | qY 
tg+n 


therefore 2 


and consequently g =; substituting for g and k, their value, in the 
equation k+tqg+n—1=s, wehavetn+m +n =s, and con- 
sequently 


Transformation of Differentials. 135 


wherefore the reduction of one differential to another will be possi- 
ble, if the difference s — m of the exponents of x out of the two 
binomials, divided by the exponent of x in the binomial, gives a posi- 
tive whole number, we then suppose, in the original equation, 
at dace (a + bx \P 

=fathe et (Axmtl4 Bamtntig Camtent. 

ek aie ae Paut—th)+ fRamda(a+ ban) ; 
and, in order to determine the coefficients, 4, B, C, P, R, &c. after 
having differentiated, divided by (a+ 6a” )¢ dx, and performed the 
operations which are indicated, we transpose the whole to one side 
of the equation, and make the sum of the quantities, which multiply 
each power of x, equal to zero, which will give as many equations as 
there are undetermined coefficients. 

129. But if we pay attention, we shall find, that when 


| at dx(a-+ ban je 


depends on I amd x(a-+ bx"), reciprocally, the latter depends on 


the former ; now, proceeding as above to reduce , 
forda(at bane to fa du(a+ bay, 
we should find that 
m—s 
n 
and that we must suppose 
foradx(a + ban )e 
= (a+ banyett (dat? 4 Bustatt 4 &. + Pam—nt1) 
+f Ras dzx(a+ba\, 


therefore, whether s be greater or less than m, provided that 


=a positive whole number, 


S—— 1 
F) 


tt —S 


or , gives a positive whole number, we may always reduce one 


of ‘these differentials to the other by substituting for the first expo- 
nent of x in the series 2x* + Bukt+4, &c. the least of the two ex- 
ponents m and s, increased by unity, and taking for g the exponent 
of x in the binomial. 

For example, if we wish to reduce 


A 2 

a* d x(b? — x?)* to dx (b? —x*)*, 
which depends on the quadrature of the circle; we see that s— m 
is here 4 — 0), which, being divided by n, that is, by 2, gives 2, a whole 
number; the reduction is therefore possible; and since the formula 


S—— mM 
t= 


—1 gives {= 1, and as moreover the least exponent 


m = O, we substitute 1 for k; we then make 


ae 


136 —Antegral Calculus. 


fod da(b% —02)F = (09 — 29) (Ae Bas) + fRdx(b2 — 2), 


1 i P 
differentiating, dividing by (b? — *)* dx, and transposing, we have 
O=Ah?— Ax? —3 But 
+ R +3 Bb? 1? —x* 

— 3423 —3 Bx*, 
whence we deduce | 
—6B—1=0, —444+5 Bb? =0, 14b?+R=0, 
wherefore, 
B=—4, =—1bb, R= {b*; 
therefore 
1 
f= d a (b? —ax* 
1 ike Oaee 

— (b? — x7)? (—i b3 x—i x) + 4 b+ fda /(b2 —2x2) + C. 

It is therefore easy, by this method, to find the differentials which 
are referred to a given differential, and consequently those which are 
referred to the quadrature of the circle, of the ellipse, and the hy- 
perbola, differentials of which it is easy to find the different expres- 
sions, by means of the different equations of these curves. 

130. We may here take occasion t» observe, that this method 
shows also the binomial differentials which are integrable ; indeed, to 
find, among such binomial differentials, as h.v* d x(a + bx” \? , those 
which are integrable, is to find those which depend on 

Rat—ldx(a+ba"), 
which has been found (90) to be directly integrable ; now it results 
from what is laid down (128) that "re 


whole number, that is, that 
s+1 
n 
which agrees with what is said (91). 

131. Let us now suppose that the two binomials which enter into the 
differentials in question, have different exponents, so that the propos- 
ed differential ish «* dx (a+ bx"), and that, to whose form we wish 
to reduce it, isa™da(a-+ ba"), p having a numerical yalue less 
than r. If ris positive, we change the differential 

hatd«(a-+ba") intoha® dz(a+ bat)r-6 x (a + bat yp. 
Then if r—p is a positive whole number, we may reduce 

hada (a+ba*)r—? (a+ba")p 
to a series of terms of this form 
(A'at + Blartn 4. Crast2r 4 &c.) dx (at bar yp, 
of which each may be reduced to the form «” da (a + ba" p by the 
preceding method, if s— m can be divided by 7, and, to reduce the 
whole to that form, we must follow exactly the method there given, 
taking for the quantity which was there called s, the greatest expo- 


must be a positive 


= a positive whole number, 


Transformation of Differentials. 137 


nent of x in the expanded value of 
hat da(a-+- bx)r-P, 
If we had, for example, 


furde (b* — x2) to reduce to fd a (b? —x*)?, 
: 3 
we should change J a? d x (b® — x?)* into 
f2? da (2 —24) (09 — 22) ot f Giada —at da) (24) 


then what we have to take for s,is 4. We suppose, therefore, con- 
formably to the method, 


pede —xtda) (b2 — a:2)2 


3 i 
= (b? — x2)? (4x + Bas) + f Rdx (b?>— x)? 

If, on the contrary, the value of r is negative, the differential, to 
‘which the proposed differential is to be referred, must be prepared 
thus, rd x(a + ba" )P—" x (a +ba"\?, then, if p—r isa whole 
number, as it will necessarily be positive (since we suppose r nega- 
tive and greater than p, whatever be the value of p) we may reduce 
amdx (a+ ba")P—"(a+bx") to a finite series of terms of this 
form, (4x + Blamt + Camt22 4 &c.) (a+ ba"), we may then 
proceed as if it were required to reduce this last to the form 
atdx(a+ba")r, that is, we may proceed in a manner precisely 
similar to that to be tollowed in the case when r is positive. 

It it were required, for example, to reduce 

dx 
a2 + 3’ 
which (110) is integrated by means of an arc of a circle, of which x 
is the tangent, and a the radius, we should change d x (a? + x?)—* 
into (a* + 27) dx (u* + x?)—*, and, as the least exponent out of the 
proposed binomial is 2, we should suppose 


fRw@ + x?) dx(a? + x*)-? 
= (a? +2?)-} (Ma-1 4+ Ba) +f ga-? dx (a* +22)-2, 


And we should proceed as above to determine the coefficients, 4, B, 
and R. ‘Then, by transposition, we should have the value of 
fer rde (a ot), 
in which we should afterwards reduce 
R (a? + x?) d x(a? + 2?)-? to Rd x (a? + 23)-?, 


ga-td ax (a? + x?)—-2 toda (a* + «*)—tor 


On Rational Fractions. 


132, Every rational differential fraction is always integrable either 
algebraically, er by arcs of a circle, or logarithms, or all these means: 
united, or by only two of them. 

They are always integrable algebraically, when they have no vari- 
ble denominator unless it be a simple quantity, excepting only the case 

18 


138 Integral Calculus. 


in which the denominator is raised no higher than to the first power, 
as we have already seen (82) 

It remains for us therefore to show the truth of our assertion, in 
the other cases; that is, when the proposed differential has a rational 
complex denominator. 

We shall suppose that the variable in the numerator of the propos- 
ed differential fraction is of a lower degree than in the denominator. 
If this were not the case, we should inake it so, by dividing the nu- 
merator by the denominator until the remaining power should be less 
than in the denominator. For example, if we had to integrate 

xe die 


lapis ar we should begin by dividing 23d a by a?+3 a x+27; 
we should find xd <x for a quotient and —Saa?d x—v2 xd x for 
a remainder ; we should again divide this remainder by the same de- 
nominator, and find —3 ad x fora quotient, and 
+8a2xdx4+3a3de 
Leis 
az+3ax+2 
Sartxdxt3atdzx 
xrdx—3adxrt+ Badin adi APP 8. 
a" +oa@a +2 

In order to discover by what means we may integrate rational dif- 
ferential fractions, we recollect that the differential of the logarithm 
of a quantity, being the differential of that quantity divided by the 
quantity itself, that is, being always a fraction, it 1s very natural to 
suspect that the integration of rational fractions often depends on 
logarithms. Let us take, for example, 


Q2al(a+ax)—2al(2Qa+2x); 
differentiating, we have 
2ada Qadx 
a-e 2ata” 
or, reducing to the same denominator, 


eM ant Meg 
Q2a*7+3aK0 + x07" 
Now, it is evident that, in order to integrate this fraction, we have 
only to decompose it into two fractions, one of which shall have 
a +«x,and the other 2a+ x for a denominator. The numerators 
will be constant numbers multiplied by da; these two fractions 
would then be integrated by logarithms. 

153, it is therefore very natural to endeavour, in order to integrate 
fractions of this kind, to decompose them into as many simple frac- 
tions as the denominator has factors, each one of which shali have 
for its denominator one of these factors. This, indeed, is the meth- 
od which we may and must pursue, when all the factors of which the 
denominator may have been formed are unequal. 

iS4. But when, among the factors of the denominator there are 
any which are equal tu each other, then we are not to expect that 
the method should be successful, because the integral cannot en- 


for a remainder; then instead of >» we should take 


Rational Fractions. 139 


dx 
(@+2)?> 
whose denominator has two equal factors a + x and a + a, we should 
find (88) that the integral of this quantity or of its equal 

dx(a+.x)~? is —(a+a)-1+ C, 
which does not depend on logarithms. But we see, at the same time, 
that if we should differentiate such a quantity as 


tirely depend on logarithms. If, for example, we had, 


—_ + 2al(a+a)4+2alQa+a)—al (Sa+a), 
we should have 
— Aad i chang 2ada , ada 
(a + 4)? ata Qatz 8a+tx’? 
or 
(Qax-+a*)dx,.:,.2ada ad«x 
(a + x)? Sak oa! 


or, reducing the whole to a common denominator, 
l0atdx+2G6aixadx+17a@ar*dr+3axidx |. 
(4+ a)? (2Qa+4+x)Ga+a) ; 
a fraction which, in order to be integrated, would only require to be 
reduced to 


Qax+ a2 if 2Qadx ada 
(@ens 7 gabe Sapa’ 


that is, to be decomposed into three fractions, of which the first 
should have for its denominator all the equal factors, and in its nu- 
merator all the powers of x less than the highest power of the de- 
nominator. .The other two fractions should have each, for its denomi- 
nator, one of the unequal factors, and no power of x in its numera- 
tor. In this manner every rational fraction may be integrated ; and 
we proceed in this manner, at least when there are no imaginary 
factors in the denominator; which case will be examined hereafter. 


Thus Ca. bo nO eke 1). die 
(M+ Na+ But? +.... Lx") 
ral, any rational fraction ; if we suppose the denominator to have a 
number m of factors, equal to # + ga number p of factors, equal to 
x +h, &c. and any number of unequal factors, represented by x + 7, 
xz -—+q,x +r, &c. the proposed fraction will be 
(at+tba+ca?+....ka—1) da 
(a+ g)™(~ + hie + &e. (w+ 7) (a +9) (w@ +1) &e. 

In order to integrate this fraction, we must suppose it equal to 
Ax™—!dx+ Bam-2dx...4+9Rdx 


, representing, in gene- 


(2 - gym 
Aw dat Bw-2de+t...Rdx 
-+ Pe SC Cy ee es &c. 
, Lede Mado | Nida ge 


er oe ee 


140 Integral Calculus. 


A, B, C, &c. being constant and undetermined coefficients. If, then, 
we can by any means determine these coeflicients, it will be easy to 
find the integral. This is evident in the case of the simple fractions 


Ldx Mdwx Nde 
t+l?>ax+tqatr 
Ll (a+), Ml («+ q), Wl (x +r), &c. As to the fraction 

As ada + ba ~*dat ... Rid x: 

(m-++2)™ : 
we take, for the sake of greater simplicity, « + g = z, which gives 
xr=z—g,anddx=dz. Bysubstituting these values, we reduce 
the whole to a series of simple quantities easily integrated. one only 


, &c. of which the integral is 


of which will have the form Ed, that is, be integrated by logarithms. 


In like manner, for the terms 


f’gP 1 Dee ARE chee Ee 
ee we take fh =z". 
ri 4 


There thus remain only two things to examine; the first is how 
to find the factors of the denominator of the proposed differential 
fraction; the second, how to find the undetermined coefficients. 

155. To find the factors of the denominator, we proceed as we 
should to resolve the equation produced by setting the denominator 
equal to zero, since (4lg. 184) to resolye an equation is to find the 
binomial factors, by the multiplication of which the equation was 
formed. ‘Thus we must employ the methods given (Alg 185, &c.), 

136. As to the manner of finding the coefficients 4, B, C, the way 
which offers itself as most natural is to reduce all the fractions in 
which they occur to the same denominator, then both members of the 
equation formed of the proposed fraction and these new fractions, 
having the same denominator, we may suppress this denominator, and 
having transposed the whole to one side of the equation, we shall find, 
that, in order that the equation should be true, independently of any 
value of x, the sum of the factors, which multiply each power of x 
must be equal to zero. ‘This condition will give as many equations 
as there are undetermined coefficients, and by means of them the co- 
efficients may be determined. ‘The following are some examples. 


t : dx 
Let it be proposed to integrate aaa ga > We suppose - 
Ade. Ke Bde 


2 
_ + ———, since the two factors of the denominator are 
- aA+- & am NX 


a+ x and a —~2z, then reducing to the same denominator, we have 


— 


az = cones eae aha suppressing the common 
a2 —— ve Sa 


denominator, dividing by da, and transposing, we have 


I1+A4z 
—Ba 


wherefore 1 — 1a—Ba= 0, and 42— Bx =0; from the last of 
these we have 4 == B, wherefore, the first becomes 


Rational Fractions, 141 


1—fAa—Aa =0, or 1—2 Aa= 6, 


1 1 
whence 4 = ——, and B = ——-; we have therefore 
2a 2a 


1 1 
Pa So en S 
a—xv? atx’ a—x 
of which the aa is 
d x I ,@ as Bo 
Let us inka 42 a second eee the fraction 
10at+da+ 26a% rdx +1702 22 dx+38axridx 
(a+ x)* (2a+42) (3a + 2) ‘ 
which was found (134) by differentiating , ‘ + 2al (a+ x) 


+2al(2a+2)—al(3a +2), we shall then suppose 
ldatdx+%6ai3xrdx+17a8x*dr+3ax8dz 
— (a$z2)? (2Qa+2) (38ap-z) 
_ (Ax + B)dx Cdx | Ddx 
tila we)? + Q2a+ex + 8a+2’ 
reducing to the same denominator, suppressing the common denomi- 
nator, dividing by d x, and transposing, we have 


10 a4 + 2a% a + 17a? x? + 3ax3 
—6Ba*—5 Bax — Br? —Axs 
—3 0a —64a2 x4 —5 Aac* — Cx = 0 
— 2Dai—7 Catx—5 Cazx*—Dex3 | 
—5 Dar*x—4Dax* 
therefore 
$a—A—C—D=0, 17a? —B—5 la—5 Ca—4Da=0, 
26 a’ —5 Ba—6 Ja*—7 Ca? —5 Da? —0, 
10a*—6 Ba? — 3 Ca*—2 Da* — 0, 
equations from which we deducet 
A=2a, B=a?, C=2a, D=—a; 


t These values are found in the following manner. 
B=170?—150?+5 €a+5Da—5 Ca—4Da 


B=2a*+Da 

26a8 —5 Ba—6Aa? —5 Da” 
ae ee ee 
Ox 26 aS—10 a8—5 Da®— 18 a8+6Ca*+6 Da*—5 Da? 


7 a* 
7C=—2a+6C0—4D 
C=—~—2a—4D 


10 a4 — 6 Bat—s Ca 
D = ORE). Itt Gin aR Gah ue at 
2 


142 Integral Calculus. 


the proposed differential is therefore changed inte 


Car+a2)dx Qadx adx 
(@ + x)* Demin: <8 @ake 


precisely as we found it above ‘The two last terms have evidently 
for their integral 2a1(2a+2)—al(3a+.); with regard to the 
Qax+ a? 


term —-——-.,— dx, we make a+ x4 =2, and have x=z—4, 
(a + x) 
and daw = dz; whence we have 
(Laz—a’*)dz Qadz a? dz 
oS REO or “ex 5 —— = 3 


2 2 
of which the integral is 2alz + as or 2al (a+ 2) + ee the 


aa 
a+a 


whole integral is +2al(a+x)4+2al(Qa+x)—al(3a+z), 
as it should be. | 

137. ‘Vius method is general. But there are several shorter ways 
of tinding the coefficients. We may, for example, find the coeffi- 
cients of simple fractions, independently of each other, in the fol- 


x a 
Jowing manner. Let 


be the fraction proposed ; h.« + a, one 


M 
of the factors of the denominator ; let P represent the other factors, 
or be the quotient of MM divided by hax —a. Conceive a to be 
Adx I: 
decomposed into Es a al ; we shall have 
ha +a P 
Ndxex  Adx Qdx fs Vie ie A Qa 
Ry) me pees eee Pig BP ek het aap 
thorefore, by reducing to a common denominator, observing that 
AE 
P= ep or P X (hav +a) = WM, 


we shall have VW== 4P + Q(hux +a). But if we differentiate the 
equation (hz + a) P=, wehaveh Pda + (ha 4 aj)dP =d M. 
Now as this equation and the equation W= AP + Q (haw + a) must 
be true for every value of «, they must be true, if we give to ~ any 
value whatever. We therefore give to x the value which gives the 


a 
most simple result, that is, the value — 7 obtained by supposing the 


_ 10a4— 1204 —6 Dad +604 412 Da? 


248 
=5a—6e—3D438a+6D.0.—2a=2D 
Os — Void ee a ae ea 


B=2a?+4+ Da=2a?— 2 = a" 
A=$a—-C—D=3a—2a+a—=24a; 


Rational Fractions. 143 


denominator hxa+a =0. Wethenhaveh Pd x =d M,and V=AP. 


dA i 
Substituting in the second the value P = a, given by the first, 


hNdzx 


d M 
one of the simple fractions, we must divide the numerator Wd x of 
the proposed fraction by d M the differential of its denominator, and 
having substituted for « the value obtained by making the denomina- 
tor of the simple fraction equal to zero, multiply the whole by the 
coefficient of x in this denominator. 

To obtain. for example, the value of the numerators 4 and B of 


; . d : 3 
the fractions eee and ee into which we above resolved the 


v 


and we have 4 = ; that is, to obtain the numerator 4 of any 


d x . .e x bl 
7a We differentiate the denominator a? — x”, which 


L een 


fraction 


gives —2adx. We then divide the numerator dx of the proposed 
. ‘ areas ; 5 
fraction by —2«d x, which gives — sy 2 which successively sub- 


stituting for z, — a and a, (which are the values obtained by making 
the denominators a + x and a—w of the partial fractions succes- 
sively equal to zero), and multiplying by 1 and — 1, the values of h, 


~ we have a and a for the values of .4 and B, as was found above. 


We might also find general rules for ascertaining the coefficients 
of the numerators of the partial fractions which have for their de- 
nominator the product of the equal roots; but shall not now stop to 
investigate them. 

138. Although the rules just given for integrating rational frac- 
tions be general, yet when some of the factors of the denominator are 
imaginary, we have, fur an integral, quantities composed of imaginary 
ones. Such an integral. is not the less real, though it is sometimes 
with difficulty reduced to a real form. In this case, we first take out 
all the real factors of the denominator, and then decompose the re- 
mainder into factors not of the first, but of the second degree, which 
are always real. ‘Then, for each factor of the second degree, 
which may always be represented by aa?+ba-+ c, we form 


Axdx+Bdzx 


—__—__—.———-, and’ determine the coeffi- 
ac*+ba+e’ 


a fraction of this form 


cients as above. 

139. If among the factors of the second degree, there are found 
any which are equal to each other, we form, for each group of these 
equal factors, a fraction of the form 

AxM@—ldoe t+ Bar *—2*de+t..i¢Qdzx 
(au* + bx” + 0)" ‘ 
n being the number of equal factors az* +ba +e. 

140. It only remains to show how these quantities may be inte- 

rated. 

With regard to the first, let us suppose, for the sake of making the 
operations more simple, that the partial fraction is reduced to the 


144 Integral Calculus. 


ANadaxrt Bd« 
terms by a. : 

We then cause the second term of the denominator to disappear by 
making x +4a’=2z; which gives x =z—ta’,and dx =dz; 
by substituting these values we obtain a quantity of the form 
Czdz+Ddz 


zz+qq ’ 


Czd 


of which, the first part Sere is integrated by logarithms (124) 


and the second by means of an arc of a circle, whose radius is g and 
tangent z. 
As to the quantities which have the form 


Ax?9-l dat Bot —-2da+.... Qdx 
we make the second term of the denominator disappear, and obtain 
a quantity of the form 
Mz »-ldz + Nxz2*"-3dz+... dz 
Ao tag HERE LS PET I PO OES. om 


which is integrated by reducing to the form —, by the meth- 


form hich may be always done by dividing both 


od given (131), the integral of the sum of the terms, in which z has 
even exponents. Those whose exponent is odd may be integrated by 
za x 


what is given in article 91, or by being reduced to , -————,,_ ac- 
eggs erie (2 %+99)” 
cording to the method given (130). 
Thus every rational fraction is either integrated exactly, or de- 


pends only on arcs of a circle or logarithins. 


On certain Transformations, by which the integration may be 
facilitated. 


141. On this subject no general rules can be given. The inspec- 
tion of the quantities, experience, and practical address will dictate, 
on each occasion, what is best to be done. 

The object of the transformations here spoken of is to render the 
proposed differentials rational, as we then know how to integrate 
them. We subjoin, however, a few observations. 

142. If .there are no radical quantities but such as are simple 
quantities, we first give them fractional exponents, which we reduce 


to the same denominator. Then, if «7 represent one of these quan- 
1 


tities so prepared, we make at =z, which gives c= z/, and 
dx=Ilz'-'dz. We substitute these values, and obtain a quantity 
entirely rational. If we have, for example, 


dua/xtadzx 
Una dia 


Rational Fractions. 145 


xtdxetadx 
xe + x? 
cng pages 


s 3 
xt + at 


we give it the form , which we change into 


Then making as = x, we have « = x°, da== 6x* dz, and conse- 
quently 
6z§dz+6az' dz 
PRET NTS 1 IN 
which is reduced to 
6z5dz+6az*dz 
z+ ' 
and easily integrated by the rules already given for rational frac- 
tions. 

143. Every quantity, in which there is only a complex radical not 
exceeding the second degree, and in which the variable under the 
radical sign does not exceed the second degree, may always be ren- 
dered rational by one or other of the two following methods: 1. Af- 
ter having freed from the radical sign the square of the variable under 
the radical sign, we make this radical equal to the same variable 

lus or minus another variable. 2. We decompose the quantity 
affected by the radical sign into its two factors, and make it, after 
being reduced to this form, equal to one of its factors multiplied by 
a new variable. 


dx : 
If we had, for example, —————, we might make ./z2 —a2 =a«4#—2; 
/ x2 — a2 
pe Yad Wap 2 __ a2) d 
then x = + » Whencedx = th Seta ban and 
2z oe 
a? — x (z* — a?) 
Mea raat me ag a 
dx d eras Big. 
whence Waa = —, which is easily integrated. 


We might also, in this same example, make 
/x2 —ad; Or /(@—a) (« + a) = (4 — a) x5 then, squaring, and 


pias a+ax* 
dividing by x—a, x-++ a =(x—a)x*; whence x= <t 
Qaz —4dazdz 
72 at == ———-3 dx = ————__ 5 
/ x2 — az pepe a x fg aaa therefore 
dx — 2dz 7 
J/z?—at ~e—1? 


which is integrated by the rules given above for rational fractions. 
These methods may be applied to the rectification of the parabola, 
of which the element Vaz? aya is Nady? ae or 


19 


146 Integral Calculus. 


We first free y? of its factors by writing it =H Je + y?3 and 


then make ae +y%=my+z, 


144. When there is no radical but a square root, and no powers 
of « but even powers, we make the radical equal to a new variable 


multiplied by the given variable. If, for example, we had weet 

é a4—wZ 
we micht make /a3—2z2 =z. If there were a second term under 
the radical sign, we might, notwithstanding, make use of this trans- 
formation, having first made the second term disappear, at least 
when there is no power of x without the radical sign. 

145. Finally, we may, with a view to making a quantity rational, 
put the variable or any fraction of the variable equal to a new varia- 
bie or some fraction of one, in which we leave something indetermi- 
nate, which may serve to efiect the object in view. For example, to 
ascertain in what case we might render the quantity 

x ™ dx (a+b ax" Pe 


rational, we should make (a + ba” )* = 29, q being indeterminate. 


ths 
if zP —a 
We should have a + b2® =z? 3 a® = ra te 
g L q m 
ZP mmm UND i. zP—a ne 
rine b : HS 6 =( b 3 
gq 1 
Gere —_ —--—1 
dont ant Jeter ad Wey 
np b ? 
therefore 
qg ws vibe Mea 
aa eke aang oat 
amda (a+ bat) = —1_ xb dx(= —y see 
np b b 5) 
m+1 


which is integrable, whatever may be the value of g, when —1 


is a positive whole number or zero (85) and which may be rendered 


m --1 


rational by making ¢ = p, when — 1 is a negative whole 


. . k ° 
number. And if the value of pis + —, & being an odd whole num- © 
ber, we may reduce it to the case mentioned (143), by making g = k, 
-- m+ k! 
if a has for its value = —, k’ being an odd whole number. 
a 


146. We shall extend transformations of this kind no farther. 


Integration of Exponential Quantities. 147 


We shall only observe that certain integrations are often facilitated 
: 1 
by making the variable equal to a fraction, such as a For exam- 


wttda +adx 
q204f ais 
—~—~23dz—azi%dz 


1l+22z 
. BY: f f Adz 
of simple quantities, and a quantity of the form i+ 


‘ 1 
ple, if we had 3 by making # = =? we should have 


, which may be reduced by division to a series 


of which we 


f) 
zz 


already know the integral. 


On the Integration of Exponential Quantities. 


147. There are no other rules to be given on the differentiation of 
these quantities, except that we may endeavour to decompose them in- 
to two factors, one of which shall be the differential of the logarithm 
of the other, or a constant part of it (27); we then divide by the dif- 
ferential of the logarithm of this second factor. ‘Thus we see that 


xy (a yla + gee ) is integrable, because the factor d yl x -+ J : ca 


is the differential of yl x, the logarithm of z¥ ; we therefore have, as 
yd x 
dylx + -— 


x 


the integral, «9 Ser oa AL ETA + 0; that is 


or zy + C. By the same rule, we see that d x e** is integrable, be 
cause d x is the differential of the logarithm of e*”, divided by a cons 
stant quantity. We have therefore 
q dx etx eir 
C* ‘= —— é 
J wi adzle ale 


~ When e is the number whose logarithm is 1, the rule is reduced to 
dividing the proposed differential by the differential of the exponent 


of e. 
If we had x d xe** to integrate, e being the number whose loga- 


rithm is 1, we might do it, when m is a positive whole number, by 
making ; 
“om do et* = e@© (Aam+ Bam™—-t4 Ham—2+ &c.+ k). 
If, for example, we have x? d x e**, we suppose 
xo? dv ete = et* (Aa* + Bu-+ E). 
Differentiating (27), and then dividing by d x e**, we have 


42 pak; 
ba age 


148 Integral Calculus. 


gx AMavtt+aBsctak 


+2A42¢+ 86; 
whence fAa=1, aB+242=0, cE+B=0; that is, 
1 —2 Q 
ai ag Ares Bie rs in ye? 


wherefore the integral Jf “2d x e8* is 
2 
ere cals inp =;) + C. 


The number e, whose logarithm is 1, may be employed with ad- 
vantage for the integration of many quantities, especially when they 
contain logarithms. For example, if we had to integrate a®dz (l2)™, 
we should make la=z=zle; wherefore xr = e* ; dxr=dzée ; 
and consequently 2” dz (lx)™ = 2™dze@t«, which is integrat- 
ed in the same case as the preceding and in the same manner. 


On the Integration of Quantities with two or more Vi arvables. 


-148. If we examine the rule given for differentiating quantities 
with several variables, we shall see, that in order to integrate diffe- 
rentials with several variables (when it is possible), we must collect 
all the terms affected by the differential of a single variable, and in- 
tegrate them as if there were no other variable but that, that is, as if 
all the others were constant. If we then differentiate this integral, 
making all the variables vary successively, and subtract the result 
from the proposed differential, the integral, thus found, is, after a 
constant quantity is added, the true integral, provided there be no re- 
mainder. If there be aremainder, it will not contain the variable, with 
reference to which the integration has been performed; we pursue 
with the remainder the same process as before, and so on with each 
variable. If, for example, we had 


Sxryda+auridy+sxcytdy+ty5dau; 
we should take the two terms affected by dz, viz. 
3a ydz+yida, 
and integrate them as if y were constant. The integral is 
at y + y? x 
Now this quantity, being differentiated with reference to x and y, 


and the result subtracted from the proposed differential, nothing re- 


mains ; we thence conclude that the integral is 73 y + y4 a + C. 
If we had 


ody+sctydrpotdz+Qezde+xdatytdys 


by collecting all the terms affected by da, and integrating, x and y 
we | 
being regarded as constant, we should have «3 y +2? z+ ee Sub- 


iracting the differential of this quantity, considered as having three 
. j oe ’ . 
variables, from the proposed differential, and there remains y? d y ; 


3 
we therefore take the integral of y3 dy, which is er , and adding it, 
re] 


together with a constant, to the integral already found, we have 


Integration of Quantities. 149 


3 3 
wytoz+— +246, 


for the integral, 

149. But as it is not always possible to integrate a differential 
with several variables, it will be well to point out a character, by 
which it may be known when it is so. 

150. 'To this end, it must be observed, that if, in any quantity Q, 
composed in any manner of two other quantities a and y, we at first 
substitute for 2, a certain quantity p, and in the result substitute q 
for y, we shall have the same final result, as if we had first substitut- 
ed gq for y, and afterwards p for x. ‘This is evident. 

151. It thence follows, that if we differentiate a quantity Q com- 
' posed of x, y, and constants, making first only x variable, and then 
differentiate the result, making only y variable, we shall have the 
same final result, as if we had first differentiated, making only y va- 
Ne and afterwards differentiated this result, making only « varia- 
ble. 

Indeed, let us conceive, that by substituting first « + d x for x, Q 
becomes Q’; we have Q’— Q for the differential. If, by substituting 
y +d y for y in this quantity, Q’ becomes Q”, and Q becomes Q’”, so 
that Q’— & becomes Q” — Q’”, we shall have Q’v— A” — V+ @Q 
for the second differential. 

Suppose now we substitute in the contrary order, and since, by 
substituting y + dy, instead of y, in &@,1t becomes Q”, we shall have 
Q” — Q for the first differential, on the supposition that y is varia- 
ble. If we now substitute x + d 7, instead of x, in this quantity, Q 
will become Q’, as above, and Q@” will become @ (150), so that 
Q'” — Q will become Q’’ — Q’, wherefore the second differential will 
be Q” — Q’— Q’” + Q, precisely the same as before. 

Let us now suppose, that 4 representing a quantity composed of 


x and y, - dy indicates the differential of 4, taken by making y 


only variable ; and dx that of 4, making x alone variable. In 


like manner, tay dx dy will indicate that 4 was first differentiat- 


ed, supposing only x variable, and the result was then differentiated, 
supposing only y variable. 

152. These explanations being thus given, let dz + Bdy be an 
exact differential, and JM its integral ; we shall therefore have 


dMdx . dMdy 


therefore 
dM d M 
IES Er ont a = B; 
therefore also 
ddM d A dd M dB 


150 Integral Calculus. 


ddM dA , dd M dB. 

dady dy?" dydz ~ da’ 

but it has just been demonstrated (151) that 
ddMdxdy  ddMdydzx 


ne 


dxdy mV eengas 


or 


heref; i ddM dd Mt oref, 1 d A _aB 
wheretore dady dydx’ elore ody ee 


that is, if 4dx«+ Bdybea complete differential, the differential of 

4 found by making only y vary and dividing by dy, must be equal 

to the differential of B found by making only « vary and dividing by 

dx. ¢ 

Thus we perceive thatiy% dx-+2y?dyis a complete differen- 

(dys) _ d (vy?) 
dy ai 


tial, because ; in fact, the first member is reduc- 


2 2 
ed to yi is and the second to asa 


dy = pe We perceive, on the contra- 


d(xy) 
dy 


ry, that xy dx-+ 2a dy is not integrable because is not equal 
d (22) 
Randa 
153. If more than two variables enter into the proposed differen- 
tial, that is, if it be of the form 
Adx+ Bdy+ Cdz, 
it is necessary, in order that it be integrable, that we have 
df dR df dC dB adc. 
iy he ES a eM es dy’ 
indeed, we may successively consider ~, y, and x as constant; and 
the differential, which has then only two terms, (since this supposi- 
tion makes either dz = 0,dx = 0, or dy = 0), must be nevertheless 
a complete differentia! since the proposed one is so. It must there- 
fore, in each of these cases, have the qualities of complete differen- 
tials with two variables 
It is easy, on the same principles, to find the necessary conditions 
of a greater number of variables. 


Remark. 


154. Let us suppose that @ is an unknown quantity composed of 
a, y, and constants, and that we know its differential .4d found by 
regarding y as constant. If we wish to find the total differential of 
Q, we suppose it to be 2d x + Bdy; then B must be such that we 
dA dB dA : 

should have = —— ; therefore d B = ——dzx ; we integrate, 
dy dx ay 

considering 2 only as variable, since x alone was made to vary in B. 


Differential Equations. 151 
dA dA 
Tg B25 Whence Bdy =dy [TE do. Now, 


since 2d x is supposed to be the differential of Q, found by making x 
vary, we have Q@ = f 4d2, the integration being performed, consider- 
ing x only as variable; therefore the complete differential of Q or of 


We have B = 


° J; : d 
fide is ida+dy a0" or the integration iy dx must 


be performed considering y as constant. 


On Differential Equations. 


155. When the proposed differential equation contains only two 
variables, x and y, and we have in one member the quantities 2 and 
d x, and in the other y and dy, the integration is reduced, for each 
member, to the rules given for differentials with a single variable. 

Thus, the equation a2” y"d x = b y@a" dy, which may represent 
any differential equation with two terms, has its indeterminate quan- 
tities separated at once by dividing by y” and by 2” , and becomes 

az™—rdx=byi-*dy, 
of which the integral is evidently 
m—7r-+-1 —n+1 
UES AUS NORE: aire 4+, 
m—r-+l g—n+t+1 

156. But as it may happen that either one or both of the members 
of the differential equation thus separated may not be capable of in- 
tegration algebraically, while the equation is nevertheless algebraical 
or may at least be reduced to an algebraical form, it will be weil to 
examine such of these cases as most frequently occur. 

If, for example, in the preceding equation, we had m—r =—1, 


and g—n=-—1, the differential equation would be reduced to 
des i . , and we can have the integral of each member only 
x 


by means of logarithms; so that we may havealax=bly+1 C4 
But this equation may be rendered algebraical, by writing it 
i == ly? -+lC or lat = Cy. 

Now it is evident that if two logarithms are equal, the quantities to 
which they belong must be equal ; wherefore 2? =: Cy? , an algebraic- 
al equation, 

157. If we had only g — n = — 1, the differential equation would 
dy 


be ax™-"daz= , of which the integral is 


a qgm—r+t 
m—r+l 
but we may give this equation an algebraical form by multiplying the 


zm hbly+l1C; 


* It is allowable to suppose that the constant quantity is a logarithm. 


152 3 Integral Calculus. 


first number by Je, e being the number whose logarithm is 1 (113) ; 
‘for we thereby produce no change in the equation. We shall then 


agm—rtl 


have 
axe 
or, (making m—u-+1l=p)le P =I1Cy*, and consequently, 
a xP 
¢ P = Cy. Hereafter we shall always indicate by e the quanti- 
ty whose logarithm is 1. 
158. Let us take as a second example the equation 


J/ ise : 

The second member expresses (110) the element of that arc of a cir- 

cle whose sine is x and radius 1. Whence z is tie sine of 46 8 ’ 
—Z23 

that is, of f ndxorn«z-++C. We have therefore, for the integral, 


zx =sin(nx-+C). In like manner, from the equation 


nd x ee 

NS a a 

J/1l—ss ; 
we should infer z = cos (nx-+ C). 
: dz 

159. In the same way, since we (110) expresses the element 

of that arc of a circle whose radius is 1, and tangent z, if we had 
dz : 
nds = fieest should conclude z = tang (nx-+ C). And if 
we had nda = ibaa ; we should, in order to reduce it to the 
a+], zz 


form of the preceding, make z = m u, m being a constant coefficient. 
bmdu ‘ 
We should then have aL fmiat? supposing therefore fm? = a, 


we should have m = ae which would give 


edu 


nN d of FN b as 
a+auy : 
whence we deduce se etoat BN dx af ; wherefore u, or ad 
se ae ee eye Oat 
1+uu b : tne 


zo = tang (Fe Saf + c). 


Therefore B= de tang (F & faf + c). 


Differential Equations. 153 


160. In the expressions sin (nx -+ C), tang (n x + €), which have 
just been found, na + C expresses the absolute length of the arc in 
parts of the radius 1. But as it is more convenient to employ the 
number of degrees, than the lengths themselves, it will be best, when 
we meet with such expressions, to estimate the arcs in degrees, 
which is easily done by dividing by the number of parts of the ra- 
dius contained ia a degree, that is, by 0,0174553 (125) or, which 
comes to the same thing, by multiplying by 57,2974166. ‘Thus the 
sine of the are whose length is 6, and the sine of the arc which has 
a number of degrees expressed by b x 57,2974166, are the same 
thing. 


161. If we had ert te — preys whose two members express 


Ji 22 /1i—y2 
the elements of two arcs which are to each other :: 1 : x, and whose 
sines are x and y; then, in order to integrate, we should make each 
member rational, by putting in the first, ./I—z2 = *./—1— 2, and 
in the second, ./T—y2 =y /—1—t. The equation would be chang- 


edt (143) into . a = te the integral of whichisulz=1t+UC, 


whence we have Cf =z"; and by substituting for ¢ and z their 
values, 


OG Ja1 — Vis) = © VET Vis), 
which expresses generally the ratio of the sines x and y of two arcs 
which are multiples one of the other. 

But in order to employ this equation, we must first determine the 
constant C. Now, supposing, as we may, that the two arcs have a com- 
mon origin, then « and y must become zero at the same time. But 
in this case, the equation becomes } 

—CVI =(—V1)*, or — C= (— 1)"; 
now (—1)" is + 1 or — I, according as 7 is even or odd; we have 
therefore —C = — 1, and C = = 1, the upper sign being for the case 
in which x is even, and the lower, when nis odd. ‘Therefore, finally 


= ly JT — Vimy) = (@ VET — Tae)". 


t By squaring, 1— «1? = —2* —2vz/—1 +%?, whence 
‘ x7 —— I} 
ia BY 
(22 +1) /—1 dz —1—27 
‘Cia — cea eee | dag 
nada m/l az 
wherefore, BC RESTELR Mo SC A aa SOD 
J/ il—x2 = 
d o1 At 
In the same manner, ee or ae, whence 
Tao 


nf—1 dz FG, GIGS | be» naz dt 
—————————— —_————, or-=-— 
x t Zz t 
20 


154: Integral Calculus. 


In each particular case, we may always make the imaginary quan- 
tities disappear ; but the simplest way will be to transpose the whole 
to one side of the equation, and make some of the real quantities — 
equal to zero; we shall then find the remaining equation to be divis- 
ible by ./—T, and that it will be the same as that formed by mak- 
ing the sum of the real quantities equal to zero. If, for example, 
we make n == 2, we shall have 

my fit fix? =? — 2/1. fi — 2 +1 —2?, 
or 
nf 1 43+ Qe%@—1 42x /—]./i— x2 —y f—1=0. 
Making then the sum of the real quantities equal to zero, we have 

/i—y2 +2x?—1L=0; 
and the whole equation is reduced to 
Qn f—1 -Sfl—2t—y/_1 =0, 
which being divided by ./—T, gives 
20/1 — 22 = Y =O, OF y = 2 0/1 — 23 3 
now, if we square this equation, and the equation 
/laays 2 eat 8) 


or rather ./) —y2 = 1—2-?, we shall have the same result. 
We may find in the same manner the cosines and cotangents of 

: ndx d 
Itiple arcs. For then we should integrate ———- = ales ADL 
multip Bee ie ety 


(111). by decomposing 1 + 2,x into (1+a./—=T ) (1—2 ./—17 ), and 
i-+ yy into(l+y /—1)(1—y /—T ); we should then finish the 
work according to the rules laid down for rational fractions (133 
& 136). 

162. While we are upon this subject, we will make known a mode 
of expressing the sine and cosine of an arc, which may be of use. 


da ; : 
Let da = / __ he the equation which expresses the relation 
9 
between anarczanditssiney. If wemake /1—yy =y /_1—z;3 


— dz dz 
we shall have dx = By ae OF se —dx./—1, of which the 


integral is lz =—a/—1+lC, orlz =—x yy le+l C(157), 
which gives z=Ce * va ; and substituting for ~ its value, we have 
y foi —VS/i—yy= Ce” ¥—1 + With regard to the constant C, 


it may be determined by observing that the arc x and its sine must be- 
come zero at the same time. We shall therefore have —1 = C; 


wherefore y /—1 — /i—yy= Bae om ; and consequently 
Vit Voi gee 5 


squaring and reducing, we have 
1—e-2tVv—! V1 tl 


BAT eT Pin fimmih 


Y= 


ie 


Differential Equations 155 


then, as y is the sine of a, we have 
ot Anes | OP at 7h hoe 
Sf 
if, in the second member of the equation Shae 
——— — , —tJf/—) 
JSJi—yy =yS/—1 +e “i , 

we substitute for y the value just found, we shall have 
pe V—hk | eV! TEE OY may 

2 oh 


pev—l TAM oa 


sinz = 


4/1 —y2 OF COS & = 


e 
C bs 


therefore for the cosine we have 
ot a7 oe p Peas Py Ea of 
ee 
But to return to the integration of equations. 
163. When the indeterminate quantities are not separate in the 
proposed differential equation, it is best, before Hao See to sepa- 


rate them, to ascertain whether the equation be not integrable in the 
state in which it is. ‘This may be known by examining (152) whether 


a; = . supposing dx 4+ Bdy =0 to represent the equa- 
tion. If this condition exists, we may integrate by the rules of 148. 
164. It may however happen that this condition does not exist, 
while the equation is nevertheless integrable ; which it may be some- 
times rendered by multiplying it by some factor composed of 2, y, 
and constants. 
Let this factor be P. Then 4P dx + BPdy =0 will be a com- 
plete differential. It is necessary therefore that 
ad(AP) _ d(BP) 
iy tray dae. 
The equation is thus reduced to finding for P a function of x, y, and 
constants, which may satisfy this equation. But as this investigation 
would lead too far, we shall only find P in the case when it contains 
only x and constants, or only y and constants. Suppose therefore 
that P is to contain only a, we shall have only 
dA dP d B 
Laalclnld SN 7 Abra aacnhl g $2 Ae 
UapR rE omy RE pau eer t 


cos © = 


whence we deduce 
d A dB 
ae 


dP dy da 

PRs Or Se men pia er? 
dA dB 
a < dx 


we shall therefore easily find P, if Cs OR RRR is reduced to a 


156 Integral Calculus. 


function of x, as is necessary in order that P should be, as we sup- 
pose it, a function of « only. 

We might also find the factor, if it were required that it should be 
composed of a function of x, multiplied or divided by a function of 
y of a known form. 

165. By this means we integrate in general any equation of the 
form ; 

Ayidy — KX yitide + Ay" dx=0, 
A, X’, X” being any functions of x; g and r any exponents. 

We might investigate whether it would not be made integrable by 
being multiplied by a factor of the form Py”, P being a function of 
x, and n and indeterminate exponent, and shouid find that this may 
be done by supposing n =—r. Butit is simpler to reduce immedi- 
ately the whole equation to the form 

yi-tdy a Fyirtt de 4 Fidz=0, 

Aiba: : AX’ AY 
by dividing by XY and y”, and representing the quotients Xx and ra 
by Fand F’’. In order then to integrate this, we suppose that P is 
the factor ; P being a function of x. We shall then have 

Pyi-'dy + FP yt-"t1dz« 4 F/Pdz=0. 
Now if P is a function of x, FP will be so likewise, and f Pi da 


will be reduced to the integration of quantities with a single varia- 
ble. It is only required then to render 
Pylic tay ot. FP yl oer oe 
a complete differential, which requires that 
EGE SE CER Gt 2) 
GR Me ae Fe of ote Page 
that is, that 


Ls Aah oe =(¢—r + 1) yt-” FP, 
whence ge aed (q—r + 1) Fadx; 
integrating, 
LP=fq—r+i1)Fdx =fq—r +1) Fdz.le, 
whence P =: efQ-+1)) Fade, 


Substituting this value of Pin the equation P y7-"dy 4 &c. and 
integrating, we have 

sei fQ@—r+) Fd + ( 
et ee meet le 1 xv q—rt+l . 9-1-1] Fdxe — 
Ppme TE + Fy duel ) +C=0. 

We have added no constant quantity in the integration of the 
equation which gives P, because, there being no condition to deter- 
mine it, we are at liberty to suppose it nothing. 

Letus take an example. Suppose that we have to integrate 

Mee 2) Br Bie 
dy + ae (bat eae iy) wie Os 


Differential Equations. 157 


Multiplying by P, we have 
d 

Pdy + pA “ 
it is then necessary that 


Ce) 
) j- x a Be 
re dy Te et 


adx 


+ P(bx? +cu4 fldz =0; 


3; whence! P=alzx or P= 2%. 


h bES 
whence =~ = 


Thus the equa ion becomes 

atdy +axt—itday 4 batt2de pcrItida 4 fatdx=0, 
of which the integral is 

F bxet3 crete fet 
Cena ge et. Fairly 

166. The general equation just integrated frequently occurs; and 
the method we have given may be applied in many other cases. 
‘The following may be useful hereafter. 

Lf we had two equations, 

dx+ady+(ba+cy) Tdt =o, 
kdx+a'dy+(b'ea+c'y) Tdt =o, 
x, y, and ¢ being variable; a, b, c, a’, &c. constant, and Tany func- 
tion of ¢, we might reduce the integration of these two equations to 
the preceding method in the followig manner. We multiply one 
of these, the first for example, by an indeterminate constant coefli- 
cient g, and adding it to the second, multiply the whole by a factor 
P, which we suppose to be a function of t; we-have 
(g P+kP)dx+(gaP-+a' P)dy 
+((¢bP+ 0 P)x+(gcP+ecP)y) Tdt=0. 

We now suppose this equation to be an exact differential. We must 
have (153) 


; d(gP+kP)_ d(((gbP+UP)x+(geP+c P)y)T) | 
iipidl Mamewne. Copia ee 


+ C=0. 


dt dy z 
: ths F EE ee ‘ACE Bani s But P being a function of 
dy dx 
t, this last equation, in which ¢ is considered constant, and therefore 
dt =0, is reduced to0 =0. And the two others give 


3d 


dP 
(g++ k) = (gb-+ 0") PT, and (Fat o) =(gc-+c’) PT; 
dP gb+b' ob gs oc ; 
whence Pee Tdt, and Chis Sasa’ dt; wherefore 


P 
putting these two values of eS 


equal to each other, and dividing by 


158 Integral Calculus. 


, / 

g ber ip a Bah © ,an equation in which g will be 
ithe oe ke gata 
of the second degree, and which, being resolved, will give two 
values of g. 

Supposing then g known, we shall easily find P, since the equa- 

{ e ! Ym t 
Hone =F Me T dt gives Pe =f Tdt. 
P ek gtk 
Now the equation (¢g P+kP)dx-+ &c. being actually an exact 
differential, if we integrate it, we shall have 
(g¢P+tkP)x+(gaP+a P)y+C=0; 

therefore if g indicates the first value of g found from the above 
equation of the second degree, and we represent by g’ the second 
value of g, and by P’, the value which P takes by substituting g’ for 
g, we shall also have (¢’ P) + k P’) «+ (g'a P’+ a’ P’)y+ C'=0, 
C’ being a new constant. Indeed, there is no reason why we should 
employ one of these values of g¢ rather than the other. And from 
these two equations it is easy to deduce the values of # and y, which 
will be expressed in terms of ¢ and constants. 

If the function 2 of t, which enters into the two equations, were 
different in each, we should proceed in the same manner, but should 
consider g as a function of ¢, and integrate as we should an equation 
of two variables, ¢ and t. 

If there were four variables, x, y, x and t, expressed by three 
equations of the form 

adatbdy+cdz+ (ext fy+thz) Tdi=0, 

and the function Z were the same in each, we should integrate in the 
same manner, by multiplying the second and third by the indeter- 
tainate constant quantities g and g¢’s we should then add these two 
products to the first equation and muitiply the whole by the factor P 
supposed to be a factor of f only. Then, supposing this new equation 
to be an exact differential, we should find (153) the equations from 
which to determine g, ¢’ and P. The equation which gives 2 or 
that which gives g’ will be of the third degree ; from which we have 
three values for ¢, three corresponding values for g’ and three for P. 
Changing the constant for each value of g, this will furnish three 
integrals, by means of which we may determine x, y and z in terms 
of We should proceed in the same way. if there were a greater 
number of variables, proyided the equations were of ithe preceding 
form. The method would be the same, if there were one or more 
terms expressed in t, dt, and constants only. 

167. And if we had in general any number m of equations com- 
prehending m -+- 1 variables, combined together in any way whatever, 
we should multiply the second, third &c. to the last, respectively by 
g, 2’, 2" &c. supposed to be indeterminate functions of these varia- 
bles, we should add them to the first equation, and multiply the whole 
by a factor P supposed to be a function’of these variables, and then 
suppose the whole equation to be a complete differential. If, for 
example, we had the two equations 


Adx+ BdytCdz=0, Mdx+ Bdy+Cdz=9), 


T dt, we have 


Differential Equations. 159 


we should multiply the second by g; adding it to the first and . 
multiplying the whole by P, we should have 


PMA+ Ag) da+P (B+ Bg)dy+P(C+ Cg) dz=0. 
Now, in order that this should be a complete differential, we must 
have (53) 


d.P(Ai+ A’'2) _ d.P(B+ Biz) 
dy TERME Si cd Big? 
d.P(A+‘Ag) d.P(C+C'g). 
Hee Fo Geo eee 
d.P(B+B'g)_d.P(C+O's) 
owiehe aye, | 
That is, 


d P : d(B + B 
TO (B+ Be) +P Cte) , 


dP d(A 4. 4 g) 
ee ae 


dP ad(C + Cg 
=> (C+Cg)+P pee OS) 


dx . 
P d(B+ Bg) dP d(c Y 
ate +B gt pont =F -(C+ C’ 2) 4 pPUEF ES) +€3) 


If, from the two last equations, we deduce values of oat and ae, 
and substitute them in the first, we shall have, after all reductions 
are made, 
is d(At Ag) d(B+ Bg) 
(C+ Og) x(t) 4 atte) 
d(B+B ge) d(Ct+C'g) 7 
eee cage ete) 


d(C +C'g) d(A oo?) 
SCR ee ey ee ee FE 0) 
( dx dz 


an equation not depending on #. We then find for g a func- 
tion of x, y, and z, the most general possible and one which 
will satisfy this equation. Having found g, we find for P a function 
of a, y and z, which shall satisfy any two of the equations first 
found above, which, indeed, often requires a great deal of research, 
but which at least is always possible. 

It is to be observed that if we had only a single equation, that is, 
if 4’ =0, B’ = 0, and C’ = 0, the last equation just found would be 
reduced to 


dA dB dB. dC q1C dA 
ap ate) ae em 8 Ge iz) =o 


which being an equation of the conditions between the coefficients 


fog: Integral Calculus. 


4, B, C, shows that in order that a differential equation with three 
variables, 4dx + Bdy + Cdz =0, may be integrable, even when 
multiplied by a factor, the coefficients 2, B, C must have the relation 
indicated by the equation ' 

d A dB 

qa pails Ta) + &e: =. 
When this condition is fulfilled, we determine the factor P, in such 
a manner as to satisfy two of the three equations 


d (AP) d(BP) d(AP) d(CP) d(BP) _ d(€P) 
Bape dhe ih hk ORL a Rh eo Salve oad yee 

We thus see what is to be done with a greater number of equa- 
tions and a greater number of variables; and we may determine, in 
the same manner, what are the equations. in which it would be suffi- 
cient that g should be a constant, or a function of one or two of the 
variables, &c. 

168. When the proposed differential equation does not come un- 
der any of the cases already given, we must endeavour to separate 
the indeterminate quantities. Sometimes the common rules of alge- 
bra are sufficient to effect this; at other times transformations are 
necessary. But there are many equations with regard to which it is 
difficult to determine what transformations are best. 

The equation aa*da+by xds yk dy(e+fx")" is sepa- 
rated immediately by division, and is the same as 


(atbyt)ardasydye+fu"), 


n kd : 
which becomes ey a aaa the integration of 


which is similar to that of a binomial quantity with a single variable. 
And if we have 
gedasaxtydyt+2abzc*ydy+ab7*yidy; 
we easily see that it may be written | 
gudxz=(x+4+2ba y? +b yt)aydy, 

which may be written 

guda=(x* +by?)? Xaydy. 
Now with a little attention we see that the separation will succeed 
if we make 2? + by? =~; indeed, we thus have a? = z— Dy” and 
vwdax=1dz—bydy; by substitution, 

tgdz—bgydy=azrydy; 
Aw dz 


a 


bg+ax y dy, which is 
8 


an equation from which we deduce 
easily integrated. 

169. As no general rules can be given for making the transforma- 
tions, we shail confine ourselves to some very general cases in which 
the separation is known to succeed. 

In general, the separation succeeds in ail those equations with two 
variables, which are homogeneous, that is, in which the indeterminate 
quantities 2 and y have, in each term, cither when combined or sepa- 
rate, the same sum of dimensions. 


Differential Equations. 161 


For, suppose 4d x + Bdy =0 to be a homogeneous equation, 
and that we divide the whole by a power of x, whose exponent is 
equal to the number of the dimensions of the equation, it is easy to 


perceive that there will remain in 4 and B only powers of & and 
constants ; so that the equation will be Fda + F’dy = 0, Fand F’ 


being functions of / and constants. This done, since 


x 
d +) xdy—yda 
- Xx 


we shall have daw = Ayam d (*) ae 7a y; if therefore we make 
ydz 


+- hab and substituting ford 2, we 
% Z z 


2 =x, we have dc = — 


Fydz Fdy 17. ‘as 
LOSE loon rcepcaones + Cue RE FP" dy = 0, Fand F’being now func- 


tions of z and constants. Now this equation gives 


hyp. 5 Fdz 
yo Fz+h 2x 


an equation entirely separated, since F and F’ contain no other vari- 
able than z. 

For example, if we had y3 da + yd y + bx3 dy =0, which is 
homogeneous, and the number of whose dimensions is 3, we should 
divide by x3, we obtain : 


4 : 1 
making, therefore, % =7,orr = Z, we should have 


zdy—ydz 
me 
substituting in the proposed equation, it becomes 
z?dy—yxzdz+zx2dy+bdy =0, 


dx = H 


d y Zaz : : : 
whence s 2xaem of which the integral is 


ly=1/1(2z%7+bd)+10C; 
whence y= C(2Qz74 b)%, or y* = C4 (2z? + B), 
: 2 
or finally y* = C4 (2 — ++ b), restoring in place of z its value ~ 


a 

170. It would be of great use to be able to render equations homo- 
geneous. But there is no general method for this purpose, and we 
are obliged to have recourse to transformations. Those which prom- 
ise any success consist in making one of the variables, or a function 
of one, or even a function of two, equal to a function of a new vari- 
able with indeterminate exponents, These exponents are afterwards 

21 


162 Integral Calculus. 


determined by the condition that the transformed equation is to be 
homogeneous. 

If, for example, we wish to find the cases in which the ‘equation 
ax™dx +b yi dy +cy*dy =0, to which form every equation of 
three terms may be reduced, may become homogeneous, we make 
x = x's; we shall then have 


ahxzmrth—-ldztbhynzitdy +cyk¥dy =0. 
Now, in order that this be homogeneous, we must have # = q h + ms 
and k= mh+h—1; whence 
n-+ i - 

LE en ee id 
m—gq-+1 m—q+l 
so that, if the exponents k, yg, m, and n are such that this last equa- 
tion may be true, we may render the equation homogeneous and 
consequently effect the separation. 

171. In general, from not having any direct methods, we endeavor | 
to reduce the proposed equations to other equations whose integra- 
tion is known. We proceed thus, for example, with the particular 
equation dy 4+ ay? da = bam d x, known by the name of the equa- 
tion of Riceati, and which cap be integrated only for certain values 
of m. 

If m were zero, it would be dy + ay? dx = bdzx, which is sepa- 
dy 
b—ay? 

But in order to integrate the equation when m has other values, 
we ust endeavor to change it into another, in which ay? and 6 
shall be multiplied by the same power of x ; it will then become sep- 
arable. ‘The following is the method by which we find the valucs of 
m which allow of this transformation. We make y = 4 x? + a%t; 
whence dy =p 4at—1dx+qaut-ttdx + x1 dt; by substitut- 
ing, we have 
pAat—-*dax + qui—ttdax +artdttax? t?dx+af? cwda 

4+ 2a A xettida =banrda. 


h= 


rable, giving = d #, which may be easil integrated. 
siving 8 


We suppose 
p—1=2p,pA+tal? =0,p+rq=q—13 g+2a1=0; 


1 
and we havep =—1,4 = id ate 2; which changes the equa- 
tion into z—*dt-azPds =bta"d a, 


or dt+tax-2tdx =ba td a, 
which will be separable if m = —4. 


z : : Lie ods 
If we make, in this last, t = zeit will be changed into 


dztbamt2z2da=—ax-2dux. 

Making then z = A’ xt’ + 2% t’, and, proceeding as before, we have 
pA’ xt da + gl att td 4 xt dll + bari tetr2t t' da 
Lb Aare tmteda top atitmt2 ida =ax-? da. 


Differential Equations. 163 


If we suppose 
2p’ +m+2=p'’—1; pd’+bA” = 0; 
qg + 2b =0, —lS=pt+ytm+2; 
we shall have 


p=—m—83, i gemaiaia qf =—2m— 6, 
and g—2m—6 dt' + ba-3m—0t tdxe=ax- dx, 
or dt’ +ba- “tt dc=axntida, 


which will be separable if —m—4—=2m + 4, orifm=— es : 
re) 


1 "og 
If we make t’ = at and afterwards z’ = 4’« +29 t”, and con- 


tinue the same process as before, we shall find successively that the 
. . 12 6 Sy 
equation is separable when m = — zm = —>, n=— et &e. 


. . mee Tee v ° e,e 
that is, in general, when m = oer hat being a positive whole num- 
C= 


ber. 
Taking the above successive substitutions for t,t’, t’’, &c. we 
shall find that y has for its expression 
y= NPA GO LaM—3 __ yy 2m—6 a 1 is 
(Aa m—5 +a m—10 1 


continuing the process until the exponent of « in the first term of the 
last denominator shall be —m+-2.r—1—13 and then the second 
term of these denominators will be s-2™—4"—?¢: ¢ being a variable 
which, after the substitution of this value of y, is determined by the 
integration of the resulting equation, which is then separable. The 
only exception is the case in which r = 1, in which we have only to 
make y = 4.x—' + 2778. 

Let us resume the equation dy + ay* dx =b«™d«x and imag- 
ine that, instead of substituting at first y = 4a? + «ft, as we have 


l 
done above, we first make y = rai and thenz=4 a? 49t, and proceed 
as above; we shall, in like manner, conclude, that we may effect the 


r : 3 
, r being a positive whole number. 


separation wherever m = Ee ey 


And the value of y will be 
1 
I= Agar m2 1 
(Amen 3 fg 1 


Continuing in the same manner until the first term in x in the last 
denominator shall be of the power —mr—2r-+13 and then the 


second term must be 2—? 4"? ¢. 
We may reduce to the same case the equation 


164 Integral Calculus. 


aidy+ay*x* da =hamda, by dividing by «2, and then mak- 
To aoe eee 

Such are the general methods to be employed when dw and dy 
do not exceed the first degree. As to equations containing different 
powers of dx and dy, as they cannot but be homogeneous with 
regard tod x and dy, we divide the whole by dz raised to a power 
equal to the sum of the dimensions of dw and dy; we then resolve 
the equation, considering 4! as the unknown quantity. Then, as 
dx and dy will not be higher than the first degree, it will be perceiy- 
ed whether the preceding methods are applicable to the equation. 


On Differential Equations of the second, third, and higher orders. 


171. The liberty we have (19) in a differentiation, of considering 
any one of the first differences as constant, will contribute in very 
many cases to facilitate the integration. But as it may happen that 
in a differentiation, we have considered as constant the differential 
not most proper to facilitate the integration ; we must begin by show- 
ing how we may reduce a differential equation in which some one 
difference is supposed constant, to another in which there shall be no 
constant. We may then suppose what we please constant. Let 
therefore Ada? + Bdady+ Cdy? + Dddy =O, be the equa- 
tion with two variables and second differences, in which the first dif- 
ference d x of one of the variabies has been supposed constant. 
After having divided this equation by dx we write it 


Adx+Bdy+ oe +Da(5")=0, 


ea ie : d 
which is in fact the saine, since, if we suppose d x constant, d ES 


is equal to ee . But if we do not consider dx as constant, then 

q dy\ _ dxaddy—dyddz | 

Peg a 
whence the equation is changed into 
Cd y? dzxddy—dyddx 
in which there is no difference constant. 
Let 
Adxi+ Bdzrdy+ Cdyrdct+Ddy3+ Ededdy+Fdyddy 
+ Gd3 y =?%, 


he an equation with third diiferences, d w being always constant. 
We divide by d x*, and have 


Cd x? d 43 dd dy dd 
Ada + Bdy+ ais +D +E +r 
ds y 
+ Gi, = 9, 


Differential Equations, 165 


which may be written 


Cdy? Ddy® dy 
Adx+ Bdy+——~— + re +Ba(5t 


dy 7 (44 1 dy 
gaps (52) T ea((s-) (=+)) pate 
and, considering every quantity as variable in the differentiations 
here indicated, we shall have an equation in which there will be no 
longer any constant differential. 
Let us apply these principles to an example. Let 

dat*dy—dy% =adaxddy+adxddy 
be an equation in which we have supposed dx constant. It cannot 
be immediately seen how this equation can be integrated ; but if we 
render ad.” variable, by writing it 


3 
dedy— Go =(ade+ 2da)a (Fe 


we can, in the differentiation indicated in the second member, con- 
sider dy as constant, and we shall have 
d ys 
dx 
which becomes, when reduced, 
dx?+uxddx+addxr—dy? =0, 
of which the integral, as may be easily perceived, is 
cdax+adx—ydy+ Cdy =0, 

adding a constant Cdy of the same order as the integral, This 
equation, integrated anew, gives 


dyddx 


dady— APNG, 


=—(adx-+xrdx) 


2 
a? +ax—- + Cy+ C= ik. 


172, Let us now examine equations with second differences and 
two variables. We give this name to those in which there is no dif- 
ference exceeding the second order, to whatever power dz and d y 
may be otherwise raised. 

We shall suppose one of the differences constant; but it will be 
easy to learn thence how to proceed if they were both variable. 

Let then 4ddy +B =0 be the general equation which may re- 
present any differential equation of the second order, with two vari- 
ables x and y, and in which dw is constant. 1 and B are functions 
of x, y,d a, dy, and constants. 

We write this equation in the form 


B—k k 


ke being an unknown function of the same nature as 4 and B. We 
then multiply by 7, supposed to be a function of 2, y, da, dy, and 
constants. We have 

Pk 


B—k 


whieh we suppose to be a complete differential. 


166 Integral Calculus. 


This done, we have three differences, viz.ddy, dy, and dx. Con- 
sidering these as the differences of so many different variables, we 
must have (153) 


eur Cay)» cam Ge) 
dy Ne ‘ ag. 


5 


aay. ddy 
B—k Pk 
iF 
a(r( dy )) re a(——) 
dx yh ay, ‘ 


From these three equations, we may, bv the process employed in 
art. 167, deduce an equation in which P shall not,occur, and which 
may serve to determine k, taking for k a function the most general 
possible of x, y, dx, and dy, with indeterminate coefficients, which 
may be substituted in this equation. After which we may determine 
P, by taking, in like manner, a function of the same kind, and such 
as to satisfy two of these three equations. But we may simplify 
this investigation, by confining it to finding for P a function of a, y, 
d x, and dy, which shall satisfy two of the equations. 

The first two of the three equations just found, give 


EA AE) ad ,, 1 d(P(B—k)) 
dy. i, agi “rah el My hun tadyn oe" 

or 

d(4P) ___ 1 1 d(PB)_ 1 + d(Pk) 

dy ee dyt > Wann dado ty edge 

d (AP) oiielen EBS) 
aa oid ook igs MCCA GT BENG 
Substituting, in the last equation but one, the value of aie de- 


duced from the last, we have 
d (AP) Whied RL a CPB)" 


J 2] 
By ay ape eid y ddy dy da” 


whence we have 


d(P B) b di(AP) » AAP) . 
P(B—k)=dy hala Dah TN Nee rorya 2 THe 
whence it will be easy to find k when P is known. 
From this last equation we deduce 
d(P?) d (AP) d (AP) 
MR y 9 1 @ OE ee SAL 2 e 
Pk=PB—dy ddy a Ob yt +dy adore 


substituting the value of P(B—k) and of Pk in the equation 


d(AP) _ 


dy * ddy 
and in the equation 


Differential Equations. 167 


P (B—k) Pk 
a( dy ALS oh gree 
dx dy 


and we shall have 


d (PB) d (AP) d(AP 
d (AP) Fara te cies 7) 
age 2 a day S 
eps PB) d (AP) d (AP) 
d m 
a( Ly Tans dx “te AY ay ) 
@ da 
(i dy d(PB) Ly d(AP) , dy? d (AF) ) 
dx dx ddy Ide ax dy 


The question is therefore reduced to finding for P a function of 2, 
y, @ x, d y, and constants, which shall satisfy these two equations. 
But although this be always possible, it is not always easy; for which 
reason, we shall leave this general investigation, and examine some 
equations more limited, but still very much extended. 

We first observe however that it is easy, by the principles just giv- 
en, to ascertain whether the equation is integrable in its present state, 
We have only to suppose P = 1, when, if the equation be integrable, 
it will satisfy the two following equations : 


dB d A dA 
iat a2 Grae ee Gays a aw) 


(Oo a ee ee ee ee 


dy ddy 
(4 ee ees 
dd dx dy 
and ax 
dB dy dB A Bee ad ye de 
tra a nae TOs de a> 
ee ee ee dy e 


This is general, whatever may be the differential equation of the 
second order, dx being constant. — 

173. Let it now be proposed to integrate the equation 

Gdx? + Hdxedy+ Kdy?+Lddy=0, 

in which the factor P which is necessary to render the equation 
integrable, need only be a function of a, y and constants. It is 
supposed, moreover, that G, H, K, and ZL, contain neither dx nor 
dy but are only functions of , y and constants. 

If we compare this equation with the general equation 

Addy + B =: 0, 

we have #=L, and B=Gdx?+Hdx«dy+Kdy?. Substi- 
tuting in the two equations found above, in order to determine P, 
and observing that we have supposed P, G, #, K and L to contain 


168 ; Integral Calculus. 


neither d x nor @ y, we shall have eS Hee K P; and a second 
‘ : A, PL). 
equation, which, after we have substituted in it for a 7 ~) its value 


K’ P, is reduced to 
d(PHdw+KPdy—de ae BCP Gdr4pdy) 


ax dy 
But since ap EL) = K° P, 
dy 
we have * 
Cag ay) a dy | ey 
d ad(KP) — Kid ie dy dx 
day: ri 4 


and consequently 


d (PL) 
d(KPdy) _ dya( d x 
dx i 
wherefore our second equation is reduced, after having divided each 
number by dz, to . 
d (PH) dd(PL) _ d(P@) 
— "dada t dy 
d (PL) 
This equation and the equation “dy = KP are the equations 
which we have now to operate on, in order to integrate the proposed 
equation. 

We observe now, that in this last equation, y only is to be consid- 
ered as variable. ‘This being fixed, performing the differentiation in- 
dicated, and deducing the value of ee we have 

i as Bh dL | 
AR aah een i 


taking therefore the integral, y only being considered as variable, 
since the differentiation was performed on that supposition, we shall 


have LP = [> dy—1L +x. 
We add the quantity /_X for a constant, by which we understand a 


dd (PL) 
dad 

to differentiate PL, making x variable, and divide afterwards by dz, 

pai ies differentiate the result, making x again variable, and divale 

by dx 


t By the expression , is to be understood that we ought 


Differential Equations. 169 


function of « and constants; because «has been supposed constant 
in the differentiation. 


é xX f aa y : 
From this equation we deduce P = ot ee i aS If we substi- 


d (2H) 


— &c., and then di- 
ax 


tute this value of P in the equation 


“K 
vide by ft Tae we shall have an equation from which to deter- 
mine X. But as X must be a function of x, it follows, that in order 
that the proposed equation be integrable by the multiplication of a 
factor composed only of x, y, and constants, all the y’s in this equa- 
tion must disappear. 
Let us suppose, for example, that we have the equation 
Qyda2 +(2 a+3ya)dedyt+2u7*dy+ 2” yddy =0, 
which in its present form is not integrable. We have 
L=2x2y, G=2y, H=22+3y2%, K=22*; 


therefore 
Shear? SRB oie aeletak ts ar 9 ry 
— By fh POH hy os yd a 
Substituting this value of P, and those of L, G, H, &c. in the equa- 
tion q se) &c., we shall have, after transposition, 
4Xy , 2y¥dX 2Xy Sy*dX  3Xy? yrddX 5 
ea Ee a cada 2 x dx we Game 


making the sum of the terms affected by y equal to zero, and then 
dividing one of the equations by y and the other by y?, we shall have, 
after making all the reductions, . 


dX 3dzx 


yp. x 


The first gives X = 2%; and this value, substituted in the second, 
satisfies it; we have therefore X = x}, and consequently 


pes fad 


Dyed 


,and —xw*ddX+3adXdx—38 Xdx? =0. 


= xY. 


If we now go back to the value of Pk, found (172), we shall have 
Pk=2ay*dr+3ux*y2dady, 
and P(B—k) =2s2% ydudy+ 22% ydy? ; 
so that the equation, brought to the general form (172), becomes 
asyr7ddy+(: a? yda+2x5ydy)dyt(2xy2dx+53 x? y2dy)dx=0. 
In order to integrate, we follow the rule given (148) ; we first take 
a? y? dd y, and consider dy only as variable, which gives x3 y? d y. 
Differentiating this quantity, considering all as variable, and sub- 
tracting from the equation, there remains 


(Qar%ydau)dy+ (Quy*dux)dx. 
We integrate the first of these terms, considering y only as variable, 
22 


170 Integral Calculus. 


and we shall have x? y? da, the differential of which, taking « and 
y as variable, subtracted from the preceding remainder, leaves noth- 
ing ; whence the integral is, when a constant is added, 

xs yrdy+a*y2?dxz+Cdx=0. 

We may take, as a second example, the equation 
2du?+(3x+4+y+42)dyda+ 2xrdy? + (a? +ay)ddy=0, 
which is integrated in the same manner. We shall find that X must 

be equal to vz, and H=cy. 
174. If, after the substitution of the value of P, in the equation 
Abie! &c., all the y’s disappear of themselves, the equation which 


must give X is then a differential of the second order; whence it 
appears, that the method is, in this case, of no use But it must be 
observed that the equation which will then be obtained, will be of the 
form 
Ada?+ BXdxz*?+ CdXdz+ Edd X =0, 

1, B. C, E being functions of x and of constant quantities. Now, 
in order to integrate this équation, we must write it thus 
AP’ dx?+BPXd2?+(C—k')Pdad t-+-k' P'd Pds+EPdd X=0. 
We now suppose that, P’ and k’ being functions of a only, the last 
four terms taken together form an exact differential ; then the first 
term, being a function of «, will be readily integrated. 

The equations which result from this supposition, are 


d (EP) d(k’ P'd X + BP'X dz) 
dx Mod ge De 
d (EP’) d((C'—K’) P'dz) 
Dp arr ion Ade ape 
d(k’ P'dX+4+BPXds) — d{(C—k) Pd X] 
d X im d x * 


and 
d[(C—k’) P'dx] b) d(BP’Xdzx) 
d x 5x aX : 


These four equations are reduced to the two following (from the 
consideration that k’, P’, 4, B, &c. do not contain P), 


j / TT SY JW , 
(EP) = k’ P,, and BP’ — HCC) 
d x dx 


ae 


Deducing from each of these equations, the value of et 
putting one.of these values equal to the other, we shall have, after 
making all reductions, 

Edk'+(0€—k)dE—K (C—k')dx+ BEdx— Ed C=0, 
a differential equation of only the first order, and on which depends 
the value of X, and consequently the integral of the proposed equa- 
tion. Supposing, therefore, that k’ has been determined by means 
of this equation, we may easily obtain P’, by means of the equation 


» and 


Differential Equations. 171 


: d(EP’) hes git Se 8 Sic 
ke! Pr = ——"7-—— » which gives ie ie Oe 


and consequently 


H k'd« 
Pose fs 


H being a constant quantity. When the values of k’ and P’ are 
found, we may find X, by substituting the values hk’ and P’ in the 
equation 
AP’ dx? + BPXd22?4+(C—k)PdxdX+kK PdXdzx 
+ EP’ddX =0, 
and integrating. Now as this equation cannot fail of being a com- 
plete differential, we have for its integral 

def APdx+XdcfBPdxe+aX fk P'dc+Lde= 0, 

L being a constant quantity. This is easily integrated by what has 
been already laid down (165). We may therefore find X whenever 
we can find k’; whence it may be laid down as universally true, that 
whenever nothing is wanting to make the equation 
Gdx2?+ Hdxdz+ Kdy? + Lddy? =0, 

an exact differential, but a factor composed of «, y, and constants, 
this equation will be always reducible to a differential equation of 
the first order, whatever may be the value of G, H, K, L. 

But if, after the substitution of the value of P, in the equation 
So &c., the equation still contains y, which cannot be made 
to disappear, without subjecting the coefficients G, H, K’, L, to cer- 
tain conditions, we conclude that the factor ? must also contain x 
and dy; we must then have recourse to the general method (172). 

We might proceed in the same manner to ascertain in what cases 
any other differential equation of the second order, of a known form, 
may be integrated by multiplication by a factor composed of «, y, 
and constants, or of x, dy, dx, and constants, or of y, d x, and con- 
stants, &c. 

175. With regard to differential equations of the third order, if 
we suppose them to be represented generally by 4d° y+ B =O, A 
and B being functions of x,y, d2,dy,ddy, and constants; if we 
suppose, moreover, that P is the factor composed of 2, y, d2,ddy, 
and constants, which will render it integrable, we may write it thus 


B—k k—h Ph 
3 —— —_—— = 
AP d3y + P Hew ddy+ P ce dy + at dx = 0. 
Then the following equations must be true, 
(B —k) ) (k —h) 
d (AP) i@ ddy 7, dar) _ iP dy /. 
ddy Th d3 y ? dy a dys ? 


wan _ $C) #° a) +) 


ea itn On Be SRS tS ce ee eh A a A SE a OES 
? 


sy dy ddy q 


172 Integral Calculus. 


(B—k) Ph 
ies ij (Gs 
dx ive diay 5 
aoe) Ph 
a(P dy a a(=~) 
» dx ae dy ; 


By means of these equations k, h, and P may be determined. But 
we shall carry this investigation no farther. 

The process would be similar for differential equations of still 
higher orders. 

176. It may be observed in conclusion, 1°. That when one of the 
two finite variables is wanting in an equation, it may be always re- 
duced to an equation of a lower degree, by making dy = pda, p 
being a new variable. 

177 2°. That the general equation 
avy tad™—lydx+bar—*ydau*+ &...+hydat+ Xda" =0, 
a, b, &c. being constants, Xa function of x and constants, and d x be- 
ing constant, may always be easily integrated by a method similar to 
that employed above, for the equation 

Adxz+ BXd«x?+CdXdx+EddX =0, 
To this end, it must be written 

Pd-y+P(a—k)d*-lydx+ Pkd*-lyda 

+ P(p—k’) a"-? ydau? + Pk'd™—2 yd x? + &e, 
He | + Phyda®+ PXdx"* =0, 

P being the factor which will render the equation integrable, and 
which we suppose to be a function of x; and k, k’, &c. indeterminate 
constants. 

We shall suppose that the terms, taken two and two, beginning with 
the first, form an exact differential. ‘This supposition will give the 
equations necessary for determining P,k,k’,&c. Having put the values 


.aP oy ive 
of me equal to each other, we shall have equations in terms of &k, 


k’, &c. by means of which & may be determined by an equation of 
the degree n. ‘The value of k being found, we may easily find that 
of k’, k”, &c. and that of / will be obtained by integrating, which 
will be without difficulty done. Then for each value of k, we shall 
have a particular integra], observing to add to each a different con- 
stant. From n—1 of these equations we may deduce the values of 
dy" —}, d y®—?, &c. and by substituting them in the last, we shall 
obtain the value of y in terms of z. 

178 3° If we should have several equations in which the differ- 
ences were not multiplied together, except that they were multi- 
plied by the constant difference, and in which the variables should 
not exceed the first degree, nor be multiplied together, we might in- 
tegrate them by multiplying the second, third. &c. each by a constant 
factor p, p', &c. adding them to the first, and multiplying the whole by 
a factor P, supposed to be a function of the variable whose difference 
is constant. We should then decompose the terms affected by the 


Differential Equations. 173 


differences of the same variable, as in the preceding equation. 
If, for example, we had 
addz+bddy+ (cdz+edy)dz+(fze+gy) dx? =0, 
and 

addz+bddyt+(cdz+e'dy)dx+(fiz+g’ y)dx? =0, 
by multiplying the second by p, adding it to the first and multiplying 
the whole by P, we should have 

Piat+ad)ddz+P(c+ecp)dzdxt+P(ftfp)zdz? 
+P(b+0'p)ddy+P(e+e'p)dydx+P(g+g'p)dx? =0. 
We should then decompose c + c’p, intoc +c’ p—k and k; and 
e+e’ palso, into e+e’ p—k’ andk’. Then supposing that the 
terms, taken two and two, form exact differentials, we should haye 
the equations necessary to determine k, k’, and P. The equation in 
terms of x will rise generally, to the degree 2m, which will furnish 
2 n integrals, by means of which we may eliminate all the differences 
and obtain the equations in terms of z and x, of y and 2, &c. 

179. 4°. If the equations were still more general, p-. p’, &c., as 
well as P, might be considered as functions of all the variables and 
their differences, and these functions might be determined by the 
condition of the total equation being a complete differential. 


CCAR ALL 
is 3 € i, ML et oo 2 DAY AB ay 
Ju ain Dita. ae “ale Nite me ts 


+ ite ee 7 
nie x Aen ot 


sy tater | \ sae : m4 
xara bad daa it 


ae r] ¥ 


es! re aS 

Bont 

ont, vanaiehee Me Me Lhd 
ae A nai Shih iat 


5 WEA 


Peas 6 


Note referred to in Art. 95. 


“ Since an equation to a conic section is always of the second de- 
gree, and since the most general equation of this degree may in 
every case be reduced to the form 
bi +cut+eu2+ft+gu+th =O), 

it follows that we may always make a conic section pass through 
five given points, provided that these points, taken three and three, 
are not in the same straight line, a conic section never meeting a 
straight line in more than two points. 

Suppose 4, B, C, D, E (fig. 56) to be five given points, having this Figs 56. 
condition. If we refer these points to the line 4D, which joins two 
of them, by drawing the lines BF, CH, EG, at a given angle or per- 
pendicular to 4D, then the distances AF, BF; AG, GE; AH, HC; 
AD, which are considered as known, may be regarded as the abcis- 
es and ordinates of a curve line. Now we may always suppose that 
this curve line has for its equation 

bi?+cut+eu2?+ft+gu+th=0; 
for, let fF =n; BF=m; AG=n; GE=7m'; AH =n’; 
CH =m’; AD=n’"’; then it is evident that, Ist, for the point A, 
we shall have u = 0, and t = 0, which reduces the equation to h = 0. 
2d. For the point B we shall have u =n, and t = m, which changes 
the equation into bm? + cmn+en?+ fm....gn=0, since h = 0. 
3d. For the point £ we shall have w= n’, and t =m’, and conse- 
quently bm’? +cmn +en2+fm+gn =0. 4th. For the point 
C, we shall in the same manner find 

bm’? + cm n’ + en"? + fm’ + gn’ =0. 
5th. For the point D, where ¢ = 0, and u =”, we shall have 
en”? ton” = 0, oren”+g¢=0. 

Now as these four equations contain all the quantities c, e, f, g, of the 
first degree, it will be easy to find their values; then, by substitut- 
ing them in the equation 

bt? +cut+eu2 + ft+guth=0, 
or rather bt? +cut+ eur + ftt+ cgu=O0, 
since h = 0, we shall have the value of c, e, f, g in quantities wholly 
known, and the equation will be divisible by 6. It will then be easy 


* 


te of the coefficients would be arbitrary; this wouid give the power 


176 Notes. 


to construct the curve and to determine whether it be an ellipse, 
hyperbola, parabola, or circle. If only four points were given, one 


of imposing, at pleasure, one condition; if only three points were 


Fig. 57. 


Fig. 58. 


given, two conditions might be imposed, and so on. 

We distinguish lines by the degree of their equation. Thus the 
straight line, whose equation is of the first degree, is a line of the 
first order. The conic sections are lines of the second order. It 
will be seen, therefore, that the above method may be used to de- 
termine the equation of a line of the third order, which may be 
made to pass through as many points less one as the general equa- 
tion of this order, with two indeterminates, has different terms. 
The same may be said of the higher orders. 

The method under consideration will serve to connect, by an ap- 
proximate and simple law, several known quantities, the law of 
which is very compounded or unknown. Suppose, for example, 
that three quantities are known, which may be represented by the 
lines CB, ED, GF (fig. 57), and that these quantities depend upon 
three others 456, 4D, .4F. It is proposed to find a quantity HT in- 
termediate between the first, or situated near them, and which is 
derived from 4H after the mannerin which CB, ED, &c. is derived 
from 4B, AD, &c. This question may be satisfied in an infinite 
number of ways by taking an equation with two indeterminates, 
u and t, having at least as many different terms as it contains such 
quantities as CB, ED, GF. But among all these different ways, 
that which is the most readily applicable to the different purposes to 
be answered by this method, is to regard the line /H as the ordinate, 
and the line 4H as the absciss passing through the given points C, 
E, G, &c., and which has for this equation 

t=a+ bu+cu? + &., 
by taking as many terms as there are quantities or points C, E, G; 
then by supposing as above, that uw is equivalent to .4B, ¢ will be 
equal to CB; and u being equivalent to 4D, ¢ will be equal to DE; 
and u being equivalent to AF, ¢ will be equal to GF, and so on; we 
have thus as many equations for determining a, b, c, &c., as we have 
points. The values of a, 6, c, &c., being determined, if we substi- 
tute them in the equationt = a+ bu + cu? + X&c., we shall have 
an equation in which every thing will be known except wu and ¢, 
accordingly if we put for wu, the known distance 4H, which answers 
to the quantity sought HJ, we shall have the corresponding value 
of ¢, or HI. If we would imitate the perimeter ABCDEF 
(fig. 58,) we should let fall perpendiculars from a certain number of 


Votes. 177 


the points of this curye upon a determinate line ATZ, since, by the 
method just laid down, we can determine the equation of a curve 
that shall pass through all these points, and in which ¢ being of the 
first degree, wu will be of the degree denoted by the number of these 
points less one; then this equation will serve to determine inter- 
mediate perpendiculars, approaching so much the nearer to the true 
ones, according as we take in the first place a greater number of 
points A, B, C, D, &e. See Bézout’s Algebra, Art. All. 


Note 2. 
General Demonstration of the Binomial Formula. ; 


In Lacroix’s Algebra, (art. 136, & seq.) is given a demonstration of 
the binomial theorem for the case of positive integral exponents. 
The following demonstration of the same formula for the case of 
exponents of any value whatever, integral or fractional, positive or 
negative, is taken from the Elements of Algebra by Bourdon, (art. 
202, & seq.) It may first be observed; that the binomial z-++a 


may be put under the form « € +5) ; Whence it follows. that 


(x + a)™=2™ CG +°) = a™ (1 +2)", supposing Se 


If then it can be shown that 7 formula 
— bg H— 2 


“(eee <1 fm tm"s za m— cog Ocha s 


holds true, whatever be aa value of m, it will follow, if we- 


a 
substitute — for z, and multiply by 2”, that 


(eae =m(1+me $n". G+...) 


m— 1 
=a" + marz™—i1-+m Bea el 


and that this last formula must be considered as true. 
Now it has been shown that when mis any whole number (Alg. 


136, p.), 
Phe Be 
(L-+2)"= Lt mz+m ~ Se z+. &c. ; 


We now proceed to inquire from what algebraical expression the 


series 
m— ai m— 


D ptved i —— = a fe, 


—,” tm 
23 


178 Notes. - 


is derived, when m is a positive fractional number P If we indi- 


cate this unknown expression by y, we have the apn 
1 m — m— 
2? +m 5 ee ys yr (1) 
If m’ be another positive fractional cc we shall in like 
manner have 


/ / eas 
y' = 1 4+ m'2z + m - 


y= l—ne tn 


= oat 


22 +. m! —_— 23+ &e. (2) 


If now we multiply i equations AG and a together, member 
by member, we shall obtain y y/ for the first member. As to the 
second member it follows from the laws of multiplication (Alg. art. 
31), that the form of a product does not depend on the particular value 
of the letters which enter into its factors ; consequently, the above pro- 
duct must have the same form as in the case where m and m’ are 
positive whole numbers. But in ig case we have 


Ltmztm~ ds alpen Niel 2 Ei 
paar a Ma ot He, 


whence 


(it mep may Tie a ..) (A+m'ztm™ z-#t+-) 
= (Ll+4+2)"7t™ =1+ (m+ m')z + (m +m’) tak Basta. 


therefore, the formula just obtained belongs equally to the case in 
which m and m/ have any value whatever, in which case we have 


y yi = 1+ (m+ m')z+ (m an mn) (mm +- 


m!' — 1) 
Let m” be a third positive fractional] eae we shall have 
yt = 1b mc bm Be? Liat 


Multiplying these equations ane member by member, we 
obtain 


yyy! V (mpm bm) (mpm mr) CEM EMD) 9 
Universally let there be a number q of see m, mm mi", 
(q being the denominator of m or P ,) we shall have 


py yee ==1-prz-r nPop . ties es (ADR 


7r representing the sum of the sll m —+- in’ _ m!! tm! 
If we now suppose m = m’ = m'=m.,..in which case we - 


Notes. 179 

haver =m+m+m+....=mq, the equation (4) becomes 
— I —1 —2 

yt =lfmg.ztmg.—t—— 2 4mg.~t— AY ay. 


Now we have by supposition, m = ’ whence m q= p; wherefore 


e shen 
agus 


ot eee, "oh hee: 

but p is a whole oy ile sO that the aot member of this equa- 
tion is the developement of (1 +- z)?; which gives us the equation 
yt = (Ll + z)?, whence y = (1 + ied =(1-+<)™; therefore we 


conclude that ’ 
m—1m—2 


(Lf2R=lpmetme a etme Bee 
m being any positive tien number wiiaeee : 

In order to demonstrate the truth of the formula for the case in 
which m is negative, either whole or fractional, it will be sufficient’ 
to suppose, in equation (3) which is formed of equtions (1) and 

(2), m= — m, which reduces equation (3) toy y = 1 (since 


1 
m-t-m'=0), from which we deduce y = ai But since, by hypothesis, 
m is negative, m’ or the value —m must be positive, and we have 
J 
also y’ = (1-+-2)™" =(1+2)—™ 
fa (chal = Oba) 
whence 
1 
OCT n)e ae (1 + z)7m ie (1 +z)”, 


and consequently 


Chey ili ny”of pels demir 


| ods ea 


Note 3. 


On the Method of Indeterminate Coefficients. From Bourdon’s 
Algebra, (Art. 208 and 209.) 


To give an idea of this method, let it be proposed to develope 


the expression in a series proceeding according to the as- 


a 
a+O0% 


cending powers of x, ‘The developement may evidently take place 


180 Votes. 


since may be reduced to the form a (@ + 6'2)—!; and by 


ere avs 
applying the binomial formula we may find the developement sought. 
Let us therefore suppose it performed, and that 


a 
eae ET a OP ea sta age ot eae A) 
A, B, C, D, E,... being functions of a, a b’, but independent of z, 
and which, as their value is to be determined, are called indetermi- 
nate coefficients. | 
In order to determine these coefficients, we multiply both mem- 


bers of the equation (1) by a’ + 6’ a, arrange the terms according to 
the powers of x, transpose the term a, and we have 


om (AetRa | +C a! ptDe|, 3 bE a rh. pases 512) 
—a+A b’ iat ae b +C b Th b! 

We now observe that, if we suppose suitable values to be given 
to A, B, C, D,:.. equation (1) must be verified, whatever value is 


given to z ; the same is true of equation (2). 
Now if we suppose z = 0, the latter equation becomes 0 = A a’ 


— a; whence we obtain-as the value of A, A =—; and if A is 


a ; 
equal to—, when z — 0, it must preserve the same value, what- 
a ’ 


ever be the value of 2, since by supposition, A is independent 
of «: thus, whatever may be the value of 2, equation (2) is reduc- 
ed to 


: / Q / “3 | 
CES Sree mie ace . T+ ++ or dividing by z 
ees Bai+Ca|(¢e«+Da “13 
=(+40+BH| +08 (3). 


Now as this equation must also be verified whatever value is 
given to x, we make 1—0, and the equation becomes B a'-A b'—0, 
A b! a b/ a b! 
ee eit gia 

As B must retain the same value, whatever may be that of z, we 
suppress in equation (3), the first term B a’ + A b’ which this value 
of B renders a to 0, and divided by x, and we have 

; o={ Ca + Dad |¢t+ Ea |e. 
+ Bb'+ Ch’) +DU 
Again making « = 9, this equation becomes Ca’ + Bb = 0, 


from which we deduce B= — 


/ ‘2 
whence we deduce C= — ta, OT babe eae. — it oi hi By 
a’ qi2 a’ a3- 
es same process. we should find Da’ +. Cb’ = 0, whence. 
Cb! ab’? b! ab? ae 
D = — —or D = — — — — 


Notes. 181 


Here it is easy to perceive that any coefficient is formed from 
/ 


that which precedes it, by multiplying that coefficient by — ee in 
this way we have 
a ae. a 0! ab’? a 63 a b/4 
= — — — te 2? — —_ 8 + — 2’... 
athe aa? a3 a? 


The fundamental principle of this method of indeterminate coef- 
ficients is this. Jf an equation of the form0 = M+ Na 4+ P 2? 
+ Qx2+...(M, N, P,... being ‘coefficients independent of x), 
may be verified, whatever value is given to x, each separate coefficient 
must necessarily be equal to 0. 

Indeed, since these coefficients are independent of 2, if we obtain 
their values, by making particular suppositions with regard to z, 
the value obtained will still belong to them, whatever value is given 
toz. Now if we make x = 0, we find M = 0, and by dividing by 
2, the equation is reduced to 

O=—=N+Pr44+Qr?4+...3 
if we make again in this new equation x = 0, we find V = 0, and, by 
dividing by z, the equation is reduced tO = P + Qa-+...&c. 
We have therefore separately _ 
M0, N=0,.P = 0, Qe 06 has 
and by this means we obtain as many equations as there are coefli- 
cients to be determined. 

The application of this method requires that one is previously 
acquainted with the mode of developing with reference to the ex- 
ponents of x. In ordinary cases the developement may proceed ac- 
cording to the different ascending powers of 2, but sometimes the 
expression must be separated into factors, before it is developed. 


The expression 


1 1 
ioe des 
1 1 1 
—-X —-—=-(A+Bre+Cr+D2-+...). 


el Bie Ore 


ay 5, for example, must be put under the form 
ee of ‘ 


, and we must then suppose 


182 Notes. 


Note 4. 


Of the Methods which preceded and in some Measure supplied the 
Place of the Infinitesimal Analysis. 


There are several methods of resolving questions analogous to 
that of the infinitesimal analysis; and although there are none which 
unite the same advantages, it may not. be the less curious to exam- 
ine the different points of view under which this theory may be 
regarded. 


~ On the Method of Exhaustions. 


This is the method which the ancients made use of in their diffi- 
cult researches, and especially in the theory of curved lines and curv- 
ed surfaces, and in the estimation of the areas and solidities contain- 
ed by them. As they admitted no demonstrations which were not per- 
fectly rigorous, they would not allow themselves to consider curves 
as polygons of a great number of sides. But when they wish to dis- 
cover the properties of a curve, they considered it as the fixed term 
or limit to which the inscribed and circumscribed polygons continu- 
ally approach, and as nearly as we please according as the number of 
sides is increased. In this way they exhausted, as it were, the space 
comprehended between these polygons and the curve; which cir- 
cumstance doubtless, procured for this mode of proceeding, the name 
of the method of exhaustions. 

As the polygons thus made use of, were known figures, their con- 
tinual approximation to the curve, so as finally to differ from it by 
less than any given quantity, led to the knowledge of the proper- 
ties of the curves under examination. 

But geometricians were not satisfied with thus inferring or divin- 
ing, as it were, the properties of curves; they would have them 
verified incontestibly ; this they effected by proving that any sup- 
position contrary to the existence of these properties led necessa- 
rily to some contradiction. ‘This kind of demonstration was called 
reductio ad absurdum. 

By this means, having first ascertained that the areas of similar 
polygons are to each other as the squares of their homologous lines, 
they inferred that circles of different radii are to each other as the 
square of their radii. This is the second proposition of the 12th book 
of Euclid, and the 287th.article of Legendre’s Geometry. Analogy led 
them to this conclusion, by imagining regular polygons of the same 
number of sides, to be inscribed in the given circles. For, as upon in- 


Notes. 183 


s 


creasing to any degree the number of these sides, their areas remain 
the squares of the radii of the circumscribed circles, they easily 
perce.ve that the same thing must hold of the circles, to which 
these polygons continually approached. But this was not enough. 
It was necessary rigorously to demonstrate that this is true in fact, 
and this they did by shewing that every contrary supposition 
necessarily leads to an absurdity. 

In this manner the ancients démonstrated that the solidities of 
spheres are to each other as the cubes of their diameters, that a 
cone is the third part of the cylinder of the same base and altitude ; 
propositions which are contained in the fourth section of Legendre’s 
Geometry. 

By means of inscribed and circumscribed figures, they also dem- 
onstrated the properties of curved surfaces and of the solidities con- 
tained by them. 'The law of continuity led them to the conclusion, 
and the conclusion was verified by a reductio ad absurdum. 

In this manner Archimedes demonstrated that the convex surface 
of a right cone is equal toa circle which has for its radius the mean 
proportional between the side of the cone and the radius of the 
base, that the whole surface of a sphere is equal to that of 
four of its great. circles, and that the surface of a spherical zone is 
equal to the circumference of a great circle multiplied by the altitude 
of the zone. . , 

It was also by a reductio ad absurdum, that the ancients extended 
to incommeusurable quantities, the relations which they had discov- 
ered between commensurable quaatities. ‘This method is certainly 
very beautiful and of very great value. It carries with it the char- 
acter of the most perfect evidence, and never permits its object to 
be lost sight of; it was the method of invention among the ancients, 
and is to this day very useful, because it exercises the judgment, 
accustoms one to rigorous exactness in demonstrations, and includes 
the germ of the infinitesimal analysis. It is true that it requires 
some considerable exertion of mind; but is not the power of pro- 
found meditation idispensabie to all those who would penetrate into 
a knowledge of the laws of nature? and is it not necessary, early to 
form the habit, provided we do not sacrifice too much time to 
its attainment ? 

On observing with attention the processes made use of in the 
methods of exhaustions, we perceive that there is a great resem- 
blance between them and those used in the infinitesimal analysis, 
In each, auxiliary quantities are employed, always containing some- 
thing arbitrary in their statement; from considering the’ properties 
of these quantities, inferences are drawn with regard to the un- 


184 Notes. 


known properties of the curve or other quantity in question. The 
auxiliary quantities are then omitted, and the desired result re- 
mains freed from every thing uncertain or arbitrary. 

But “though few things more ingenious than this method have 
been devised, and though nothing could be more conclusive than — 
the demonstrations resulting from it, yet it labored under two very 
considerable defects. In the first place, the process by which the 
demonstration was obtained was Jong and difficult ; and, in the sec- 
ond place, it was indirect, giving no insight into the principle on 
which the investigation was founded. Of consequence, it did not 
enable one to find out similar demonstrations, nor increase one’s 
power of making more discoveries of the same kind. It was a dem- 
onstration purely synthetical, and required, as all indirect reasoning 
must do, that the conclusion should be known before the reasoning 


is begun.” * 


In the hands of Newton, this doctrine made great progress 
towards perfection. His prime and ultimate ratios are precise- 
ly the ratios made known by the gradual approximation of the 
auxiliary quantities to the quantities whose properties are sought. 
By this theory Newton extended the principles of the method of 
exhaustions, and he simplified its processes by freeing it from the 
necessity of having its results verified by a reductio ad absurdum, 
and by shewing that these results are sufficiently established by 
the accuracy of the mode employed to obtain them. 

Newton thus expresses himself, in the conclusion of the view he 
gives of his theory, “These lemmas are premised to avoid the 
tediousness of deducing perplexed demonstrations ad absurdum, ac- 
cording to the methed of the ancient geometers.” t+ 

This great man advanced this doctrine far more considerably, by 
reducing this very method of prime and ultimate ratios to an algo- 
rithm, in his method of fluxions. By means of this calculus he in- 
troduced into algebraical analysis, not only these prime and ultimate 
ratios, but their terms taken separately, which was a modification of 
great importance, on account of the new means of transformation 
which it furnished. _ Newton, however, did not enjoy this glory 
alone ; he shared it with Leibnitz, who had the advantage of pub- 
lishing his algorithm first, and who, being powerfully seconded by 
other celebrated geometricians, associated with him, advanced his 
method far more rapidly than the method of Aratshe was eroHent 
forward during the same time. 


* Playfair. View of the Progress of Mathematical and Physical Science. 
+ Scholium to Lemma XI. Sec. 1, Book I. of the Principia. 


Notes. 185 


On the Method of Indivisibles. 


Cavalieri was the forerunner of the inventers of the infinitesimal 
analysis, and opened the way for them by his Geometry of Indivisibles. 
He was led to this by a tract of Kepler, on the measure of solids, in 
which he introduced, for the first time, the consideration of intinite- 
ly great and infinitely small quantities. 

In the method of indivisibles, solids are considered as composed 
of an infinite number of parallel surfaces, surfaces as composed of 
an infinite number of lines, and lines of an infinite number of points. 

These suppositions are certainly absurd, and should be employed 
with caution. They are rather to be considered as means of 
abbreviation, by the help of which, we obtain readily and easily, 
in many cases, what would require long and laborious processes by 
the method of exhaustions. For example. 

Let 4B (jig. 59,) be the diameter of a semicircle 4GB, ABFD 
the circumscribed rectangle, CG the radius perpendicular to DF ; 
draw the two diagonals CD, CF, and, through any point m of the 
straight line AD draw the straight line mnp g perpendicular to CG, 
cutting the circumference in the point , and the diagonal CD in the 
point p. 

Conceive the whole figure to turn about CG as an axis; the 
quadrant ACG will generate the solid volume of the hemisphere 
whose diameter is 4B, the rectangle DGC will generate the cir- 
cumscribed right cylinder, the right angled isosceles triangle CGD 
will generate a right cone having for its altitude and for the radius 
of its base the equal lines CG, DG ; finally, the three straight lines 
m g,ng, pg, will each generate a circle, whose centre will be at 
the point g. 

Now the first of these three circles is the elemeni of the cylinder, 
the second is the element of the hemisphere, and the third that of 
the cone. 

Moreover, the areas of these circles being as the squares of their 
radii, and these three radii being the hypothenuse and sides of a 
right-angled triangle, (since Cg = p g, and mg = Cn), it is 
evident that the first of these circles is equal to the sum of 
the two others: that is, the element of the cylinder is equal to the 
sum of the corresponding elements of the hemisphere and cone, and 
as itis the same with all the other elements, it follows that the 
total solidity of the cylinder is equal to the sum of the total solidity 
of the hemisphere and the total solidity of the cone. 

But we know that the solidity of the cone is one third of that of 


the cylinder, therefore the solidity of the hemisphere is two thirds ; 
24 


186 Notes. 


that is, the solidity of the entire sphere is two thirds of the solidity 
of the circumscribed cylinder, as was discovered by Archimedes. 

Cavalieri professed to consider his method as only a corollary of 
the method of exhaustions, but confessed that he could not give a 
rigorous demonstration of it. The great geometers who suc- 
ceeded him soon caught the spirit of this method, and it was in 
vogue with them, until the discovery of the new mode of calcula- 
tion. It was to this that Pascal. and Roberval owed the success of 
their profound researches on the cycloid. The former of these dis- 
tinguished authors thus expresses himself in relation to this subject. 

‘¢ For this reason, I shall not hesitate hereafter to make use of the 
language of indivisibles—the sum of the lines, the sum of the planes ; 
1 shail not hesitate to use the expression, the sum of the ordinates, 
which seems not to be geometrical, to those who do not understand 
the doctrine of indivisibles, and who think it is sinning against ge- 
ometry, to express a plane by an indefinite number of lines. But 
this comes of their not understanding it, since nothing is meant 
thereby, but the sum of an indefinite number of rectangles, each 
formed by an ordinate and one of the small equal portions of the 
diameter, the sum of which is certainly a plane. So that when we 
speak of the sum of an indefinite multitude of lines, we have always 
reference to a certain straight line, by the equal and indefinite por- 
tions of which they are multiplied.” 

‘This passage is remarkable, not only as it shows that these geom- 
eters knew how to appreciate rightly the merit of the method of 
indivisibles, but still more, as it proves that the notion of mathe- 
matical infinity, in the very sense which is at this day given it, was 
not unknown tothem. For it is evident, from the passage just cited, 
that Pascal attached to the word indefinite the same signification which 
we attach to the word znjinite, that he called by the word small, what 
we understand by infinetely small, and that he neglected, without 
hesitation, these small quantities by the side of finite quantities: 
for we see that he considered as simple rectangles the trapezoids 
or small portions of the area of the curve, which are comprehended 
between two consecutive ordinates, neglecting, consequently, the 
small mixtilineal triangles, which have for their bases the difference 
of these ordinates. No one, however, has dared to reproach Pascal 
with want of rigour. 

We'shall conclude the notice of this method with one or two 
examples. 

Common algebra teaches us how to find the sum of any number 
of terms taken in the series of natural numbers, the sum of their 


Notes. . 187 


squares, that of their cubes, &c. and this knowledge furnishes to the 
geometry of indivisibles, the means of estimating the area of a great 
number of rectilineal and curvilinear figures, and the solidities of a 
great number of bodies. 

, Let there be, for example, a triangle; from its vertex let falla 
perpendicular upon the base; divide this perpendicular into an in- 
finite number of equal parts, and through each of the points of di- 
vision draw a straight line parallel to the base, and terminating in 
the two other sides of the triangle. 

According to the principles of the geometry of indivisibles, we 
may consider the area of the triangle as the sum of all the parallels 
which are regarded as its elements ; now, by-a well known property 
of triangles, these straight lines are proportional to their distances 
from the vertex; therefore, the altitude being supposed to be divid- 
ed into equal parts, these parallels increase in an arithmetical pro- 
gression, of which the first term is zero. 

But in every arithmetical progression, whose first term is zere, 
the sum of all the terms is equal to the last multiplied by half the 
number of terms, Now, in this case, the sum of the terms is rep 
resented by the area of the triangle, the last term by the base, and 
the number of terms by the altitude. ‘Therefore the area of every 
triangle is equal to the product of its base by half its altitude. 

Again; let there be a pyramid: From its vertex let fall a per- 
pendicular upon the base, divide this perpendicular into an infinite 
number of equal parts, and through each point of division, let a 
plane pass parallel to the base of this pyramid. 

According to the principles of the geometry of indivisibles, the 
intersection of each of these planes, with the solidity of the pyra- 
mid will be one of the elements of this solidity, which will be 
simply the sum of all these elements. 

But by the properties of the pyramid, these elements are to each 
other as the squares of their distances from the vertex. Calling 
the base of the pyramid B, its altitude .1, any one of the elements 
just mentioned b, its distance from the vertex a, and the solidity of 
the pyramid 5, we shall have | 

fae A REO COE 
therefore 


Therefore S, which is the sum of all these elements, is equal to 


B ie 
the constant quantity qa multiplied by the sum of the squares a2 ; 


188 | Notes. 


and since the distances a increase in an arithmetical progression, (the 
first term ofwhich is zero, and the last 4,) that is as the natural num- 
bers from 0 to 2, the quantities a? will represent the squares of 
these distances from 0 to .4?. 
Now common algebra shows us that the sum of the squares of the 
natural numbers from 0 to A? inclusively is 
2A343A24 A 
Pier ere 
But the number 4 in this case being infinite, all the terms which 
follow the first in the numerator disappear, by the side of this first 
term; therefore this sum of the squares is reduced to x 3, 


Multiplying therefore this value by the constant quantity +5 


found above, we shall have for the solidity sought 

; S=1BA; 
that is, the solidity of a pyramid is the third part of the product of 
its base by its altitude. 

By a similar process it is proved that, generally, the area of any 
curve which has for its equation 

ay™=2", 
© m 
- m+n 
ing absciss, m, n, any exponents, whether integral, fractional, posi- 
tive, or negative. 

‘Thus the method of indivisibles supplies insome respects the place 
of the integral calculus; it may be regarded as corresponding to the 
integration of simple quantities, and this certainly was a great dis- 
covery for the time of Cavalieri. 


. XY; Yrepresenting the last ordinate, XY the correspond- 


On the Method of Indeterminate Quantities. 


It seems to me that Descartes, by his method of Indeterminates, 
approached very near to the infinitesimal analysis, or rather, that 
the infinitesimal analysis is only a fortunate application of the 
amethod of indeterminates. 

Let there be an equation with only two terms 

A 4 Bsz= 0, 
in which the first term is constant and the second susceptible of be- 
ing rendered as small as we please. According to what has been 
shown, (note 3) this equation cannot hold unless the terms 4 and Bx 
are each, separately, equal to zero. Therefore we may lay it down 
as a general principle, and as an immediate consequence of the 
method of indeterminates, that if the sum or difference of two 
pretended quantities is equal to zero, and vf one of the two may be sup- 


Notes. ? 189 


posed as small as we please, while the other contains nothing arbitrary, 
these two pretended quantities will be each separately equal to zero. 

This principle alone is sufficient to resolve by common algebra all 
quéstions falling under the infinitesimal analysis The respective pro- 
cesses of the two methods, simplified as they may be, are absolutely 
the same. The whole difference consists in the mode of considering 
the question. The quantities which in the one are neglected as in- 
finitely small, are understood in the other, though considered as finite, 
since it is demonstrated that they must eliminate themselves, that 
is, that they must destroy each other in the result of the calculation. 

Indeed, it is easy to see that this result can only be an equation of 
two terms, of which each is separately equal to zero. We may 
therefore beforehand, suppose to be understood in the course of the 
calculation, all those quantities which belong to that one of these 
two terms, of which no use is to be made. Let us apply this 
theory of indeterminates to some examples. 

Let it be proposed to prove that the area of a circle is equal to 
the product of its circumference by half the radius; that is, that 
calling this radius R, the ratio of the circumference to radius =, 
and consequently the circumference w R, the surface of the circle S, 

S= $ xR ?, 

In order to do this, we inscribe in the circle a regular polygon, 
then double sucessively the number of its sides, until the area of the 
polygon differs as little as we please from that of the circle. At the 
same time, the perimeter of the polygon will differ as little as we 
please from the circumference of the circle, and the straight line 
drawn from the center to the middle of a side, as little as we please 
from the radius R. Then the surface S, will differ as little as we 
please from § + R?; consequently if we make 

= 77h? +49, 
the quantity 9, if it is not zero, may at least be supposed as small 
as we please. Now we put this equation under the form 
a 2 tH) 0; 
an equation of two terms the first of which contains notaing arbitra- 
ry, while the second, on the contrary, may be supposed as small as we 
please; then, by the theory of indeterminates, each of these terms 
separately is equal to 0; thus we have 
S—i-n2R? aout). Ons oot eK? : 


which was to be demonstrated. 


ra. 


190 JVotes. 


Let it be proposed to find the Solidity of a Pyramid whose Base is B, 
and Altitude A. 


Conceive this pyramid to be divided into an infinite number of 
horizontal segments of the same thickness. Each of these segments 
may evidently be regarded as composed of two parts, of which one 
is a prism having for its base the smaller of the two bases of the 
segment, and the other is a kind of ungula surrounding this prism. 
Let x be the distance of any one of these segments from the vertex of 
the pyramid, and «’ the thickness of the segment. In the pyramid, 
the areas of the sections made parallel to the base are as the squares 
of their distances from the vertex ; consequently the superior or small- 


2 


east 
er base of the segment, at the distance « from the vertex, is Fat" 


Accordingly the solidity of this segment, without its ungula, is 
B 
ar 2? a’; therefore the total solidity of the pyramid, without the 


ungulas, is the sum of all these elements. And since x’ may be sup- 
posed as small as we please, each ungula may consequently be sup- 
posed as small as we please, relatively to the solidity of the seg- 
ment; whence the sum of all the elements differs as little as we 
please from the solidity of the pyramid sought. Calling, therefore, 
this solidity S, we have 
: B 
. Ss sum 43 2? a + 9, 

¢ designating a quantity which may be supposed as small as we 
please. 

But since B, 4, and « are constant quantities, that is, the same 
for all the segments, it is evident that 

sum —> x? x’ is the same as 


B Bea's iden Nex 
Aa iE ae Nera 2 


Now - is evidently the number of segments comprehended be- 


% = 
tween the vertex and a, therefore sum (+) for the entire pyr- 


amid, is the sum of the squares of the natural numbers from 0 


A 
to is See AL 
x 


But we know that this series of the squares of the natural numbers is 


2 48 ‘dos 
(= Teeter =); 


substituting this sum in the equation done above, we shall have 


Bini 2A rs bn A 
p= sa (= + "2 +—)+4% 


& 


Notes. 191 


or, by transforming in order to separate the arbitrary terms from 
those which are not so, / 
rhe 2 A 
(S$ B4)—(F (AS +>) +9) =, 

a perfectly exact equation of two terms, the first of which contains 
only definite quantities, or such as are not arbitrary, and the second 
may be made as small as we please. ‘Therefore, each of these 
terms taken separately is equal to zero ; whence we have from the first 
; S—fBA=O0orS=iBA, 
which was to be found. | 

The solution here given is analogous to the method of indivisi- 
bles, or rather it is the method of indivisibles rendered rigorous by 
certain slight modifications, derivedfrom the method of indertermin- 
ates. We will now apply this method to the same question, making 
use of the notation of the infinitesimal analysis, to show how these 
methods are related to each other, or rather that they are only one 
and the same method considered under different points of view. 

Retaining the denominations made use of above, we have d S for 
the element of this pyramid. We have moreover as the value of 
the same element, neglecting the ungula, 

7B Ur bE 5 
we have therefore exactly 
dS ==, dete, 

g representing a quantity which may be supposed as small as we 
please, relatively to each of the other terms. 

Taking, in each member, the exact sum of the elements, we have 
the rigorous equation 


S = sum >, x? de + sum $ Oe sheers (1) 


Now the common integral = “22 dx of the first term of the 


second member is 


saat +, 


C indicating a constant quantity ; but the exact differential of thie 
integral is not 
Aa 


A2 a 


B B 
aga ett de +4, (82dn? + dx*), 


192 Notes. 
that is, we have exactly 


B 
2 2 
a (sa o4+0)=5 Soldat ss = (3 ad 22 +d x3); 


taking then the exact sum i each 4 ee we have 


B 
(sa “i > 4 C) = sum G52? d-+sum F(3ed a? +d 2%) 


3 A? ‘: 
or, transposing, 
B B 
octets eat 
sum 73 2 iz=(sp 
Substituting in equation (1), we shall have exactly 


S = ae xs c) — (sums; (3 2 da?-+ da?)—sum °), 
an equation in which the last term only contains arbitrary quantities 
and may be supposed as small as we please. For the sake of con- 
ciseness, make this term ¢; the equation will become, by transpo- 
sition, | 


ee c) — sum es (3ada2+d x3). 


(s a (eat + c)) Lae 


an equation of which, by the principles of the method of indeter- 
minates, each term taken separately is equal to zero, whence 


S = e3 + C, 


B 
3 A? 
In order to determine C, we have only to make x = 0, then we 
have S = 0, whence C =0; wherefore the equation is reduced to 
B 
on aire.) 
that is, the solidity of the pyramid from the vertex to the altitude 


a3 


« is te Hae in order therefore to obtain the whole solidity of the 


pyramid, we have only to suppose x = 4, which will give 
S = i BA. 

This solution, as may be easily seen, is no other than that which 
would be obtained by the processes of the infinitesimal analysis, by 
neglecting nothing, and the common infinitesimal analysis is only an 
abbreviation of these processes, since we neglect only the quanti- 
ties 9, 2’, which, in the result of the calculation, fall only on that 
one of the two equations into which the quantities are decomposed, 
of which no use is made. Now, what the infinitesimal analysis neg- 
lects may, by a simple fiction under the name of quantities infinitely 
small, be understood, in order to preserve the rigour of geom- 
etry during the whole course of the calculation. We thus see that 
ihe method of indeterminates furnishes a rigorous demonstration of 


Votes. 193 


the infinitesimal calculus, and that it affords at the same time the 
means of supplying \the place of it if we choose by common algebra, 
It were to be wished, perhaps, that this course had been pursued in 
arriving at the differential and integral calculus ; it would have been 
as natural as the method that was actually taken, and would have 
prevented all difficulties. 


Of the Method of Prime and Ultimate Ratios or of Limits. 


The method of prime and ultimate ratios or of limits, has also its 
origin in the method of exhaustions, of which it is, properly speak- 
ing, only a development and simplification. We owe this useful im- 
provement to Newton, and it is in his book of the Principia that 
it is to be studied. It will be sufficient for our purpose to give 
here a succinct idea of it. 

When two quantities are supposed to approach each other con- 
tinually, so that their ratio or the quotient arising from dividing the 
one by the other differs less and less, and finally as little as we 
please, from unity, these two quantities are said to have for their 
ultimate ratio a ratio of equality. 

In all cases, when we suppose different quantities to approach 
respectively and simultaneously other quantities which are consid- 
ered as fixed, until they differ respectively and at the same time as 
little as we please, the ratios which these fixed quantities have to 
each other are the ultimate ratios of those which are supposed to 
approach them respectively and simultaneously, and these fixed 
quantities themselves are called limits or ultimate values of the quan- 
tities so approaching. 

These values and ratios are called ultimate values and ultimate 
ratios respectively or prime values and prime ratios, of the quanti- 
ties to which they are referred, according as the variables are con- 
sidered as approaching to or receding from the quantities, considered 
as fixed, which serve as their limits. 

These limits, or quantities considered as fixed, may, however, be 
variable, as would, for example, be the coordinates of a curve; 
that is, they may not be given by the conditions of the question, 
but be only determined by the subsequent hypotheses on which the 
calculation rests. Thus, for example, though the coordinates of a 
curve are comprehended among the quantities called variable, be- 
cause they are not of the number of the data; yet, if 1 propose a prob- 
lem to be resolyed respecting any particular curve, as that of draw- 
ing a tangent to it, it will be necessary, in order to establish my rea- 
sonings. and calculation, that J should begin by assigning determinate 


OF 


194 Notes. 


values to these coordinates, and that I should continue to the end of 
the process to regard them as fixed. Now these quantities con- 
sidered as fixed, are comprehended, as well as the data of the prob- 
lem, among the quantities called limits. 

These limits are the quantities whose ratio is sought. Those 
which are supposed gradually to appreach them are only auxiliary 
quantities, which are interposed to facilitate the expression of the 
conditions of the problem, but which must necessarily be eliminated 
in order to obtain the result sought. 

We thus see the analogy which must exist between the theory of 
prime and ultimate ratios and the infinitesimal method. What, in 
the latter, are called infinitely small quantities, are evidently the 
same as the difference between any quantity and its limits; and 
those quantities whose ultimate ratio is a ratio of equality, are those 
which, in the infinitesimal analysis, are said to differ infinitely little 
from each other. 

The princpal difference between that which is called the method 
of limits and the infinitesimal method, consists in this, that in the 
former, we can admit into the process of calculation only the limits 
themselves, which are always definite quantities, while in the latter 
we may also employ the variable quantities, which are supposed to 
approach them continually, as well as the difference between them 
and their limits. ‘This gives the infinitesimal method more means 
of varying its expressions and its algebraical transformations, with- 
out introducing the least difference in the rigour of the processes. 

The property thus obtained by the infinitesimal method renders 
it susceptible of a new degree of perfection still more important, 
which is the power of being reduced to a particular algorithm. 
For these differences between the variable quantities and their lim- 
its, are what we distinguish by the name of differentials of their 
limits, and the simplifications to which the admission of these quan- 
tities into the calculation gives occasion, are precisely what gives 
the infinitesimal analysis its importance. 

The method of limits or of prime and ultimate ratios is neverthe- 
less far superior, for the facility of its processes, to the simple 
method of exhaustions; since it is at least freed from the necessi- 
ty of areductio ad absurdum for each particular case, by far the 
most difficult operation in the method of exhaustions; while, by 
the other method, it is sufficient, in order to prove the equality 
of any two quantities, to show that they are both limits of the same 
third quantity. 


Notes. 195 


If this method were always as easy in its application as the com- 
mon infinitesimal analysis, it might seem preferable, for it would 
have the advantage of leading to the same results by a process 
direct and always clear. But it must be confessed that the method 
of limits is subject to a considerable difficulty which does not be- 
long to the common infinitesimal analysis. It is this, that as 
the infinitely small quantities are always connected by pairs, and 
cannot be separated from each other, we cannot introduce into the 
combinations, properties which belong to each of them separately, 
nor subject the equations in which they occur to all the transfor- 
mations which would be necessary in order to eliminate them. 


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